6.3e: Verifying Trigonometric Identities

A: Simplify

Exercise \(\PageIndex{A}  \)

\( \bigstar \\[4pt] \) Simplify each of the following to an expression involving a single trig function with no fractions.

1. \(\csc (t)\tan \left(t\right) \\[1pt] \) 2. \(\cos (t)\csc \left(t\right) \\[1pt] \) 3. \(\sin x \cos x \sec x\\[1pt] \) 4. \(\tan x\sin x+\sec x\cos^2x\\[1pt] \) 5. \(\sin(-x)\cos(-x)\csc(-x)\\[1pt] \) 6. \(\csc x+\cos x\cot(-x)\\[1pt] \) 7. \(-\tan(-x)\cot(-x)\\[1pt] \) 8. \(3\sin^3 t\csc t+\cos^2 t+2\cos(-t)\cos t\\[1pt] \) 9. \(\dfrac{\sec \left(t\right)}{\csc \left(t\right)\; } \\[4pt] \) 10. \(\dfrac{\cot \left(t\right)}{\csc \left(t\right)} \\[4pt] \) 11. \(\dfrac{\sec \left(t\right)-\cos \left(t\right)}{\sin \left(t\right)} \\[4pt] \) 12. \(\dfrac{\cot t+\tan t}{\sec (-t)}\\[4pt] \)

13. \(\dfrac{-\sin (-x)\cos x\sec x\csc x\tan x}{\cot x}\\[4pt] \) 14. \(\dfrac{\sin ^{2} \left(t\right)+\cos ^{2} \left(t\right)}{\cos ^{2} \left(t\right)} \\[4pt] \) 15. \(\dfrac{1-\sin ^{2} \left(t\right)}{\sin ^{2} \left(t\right)} \\[4pt] \) 16. \(\dfrac{1-\cos ^2 x}{\tan ^2 x}+2\sin ^2 x\\[4pt] \) 17. \(\dfrac{1+\tan ^2\theta }{\csc ^2\theta }+\sin ^2\theta +\dfrac{1}{\sec ^2 \theta }\\[4pt] \) 18. \(\dfrac{\tan \left(t\right)}{\sec \left(t\right)-\cos \left(t\right)} \\[4pt] \) 19. \(\dfrac{1+\cot \left(t\right)}{1+\tan \left(t\right)} \\[4pt] \) 20. \(\dfrac{1+\sin \left(t\right)}{1+\csc \left(t\right)} \\[4pt] \) 21. \(\left (\dfrac{\tan x}{\csc ^2 x}+\dfrac{\tan x}{\sec ^2 x} \right )\left (\dfrac{1+\tan x}{1+\cot x} \right )-\dfrac{1}{\cos ^2 x}\\[4pt] \)

Answers to odd exercises.

1. \(  \sec (t) \)     3. \(\sin x  \)     5. \(\cos x\  \)    7. \(-1  \)     9. tan(\(t   \))     11. tan(\(t  \))     
13. \( \tan^2 (x) \)      15. \( \cot^2 (t)  \)     17. \(\sec^2 (x)  \)     19. cot(\(t   \))     21. \( -1 \)

\( \bigstar \) Simplify the first trigonometric expression into an expression that is exclusively a function of the second expression.

24. \(\tan x\);  \(f( \sec x) \\[4pt] \) 25. \(\sec x\);  \(f( \cot x) \\[4pt] \) 26. \(\sec x\);  \(f( \sin x) \\[4pt] \) 27. \(\cot x\);  \(f( \sin x) \\[4pt] \) 28. \(\cot x\);  \(f( \csc x) \\[4pt] \) 29. \((\sec x+\csc x)(\sin x+\cos x)-2-\cot x\);  \(f( \tan x) \\[4pt] \) 30. \(\dfrac{\tan x+\cot x}{\csc x}\);  \(f( \cos x)  \\[4pt] \)

31. \(\dfrac{\sec x+\csc x}{1+\tan x}\);  \(f( \sin x) \\[4pt] \) 32. \(\dfrac{1}{\csc x-\sin x}\);  \(f( \sec x, \; \tan x) \\[4pt] \) 33. \(\dfrac{1}{\sin x\cos x}-\cot x\);  \(f( \cot x) \\[4pt] \) 34. \(\dfrac{\cos x}{1+\sin x}+\tan x\);  \(f( \cos x) \\[4pt] \) 35. \(\dfrac{1-\sin x}{1+\sin x}-\dfrac{1+\sin x}{1-\sin x}\);  \(f( \sec x, \; \tan x) \\[4pt] \) 36. \(\dfrac{1}{1-\cos x}-\dfrac{\cos x}{1+\cos x}\);  \(f( \csc x) \\[4pt] \)

Answers to odd exercises.

25. \( \pm  \sec  x  \)     27. \( \pm  \cot  x \)     29. \(\tan x  \)     31. \(\dfrac{1}{\sin x}  \)     33. \(\dfrac{1}{\cot x}  \)     35. \(-4\sec x \tan x  \)

B: Verify (no fractions)

Exercise \(\PageIndex{B}\)

\( \bigstar \) Verify the following identities.

41. \(\cos \theta   \tan \theta  =~ \sin\;\theta\\[4pt] \) 42. \(\sin\;\theta ~ \cot\;\theta ~=~ \cos\;\theta\\[4pt] \) 43. \(\text{sec}(a) - \text{cos}(a) = \text{sin}(a) \text{tan}(a) \\[4pt] \) 44. \(\cos x-\cos^3x=\cos x \sin^2 x\\[4pt] \) 45. \(\tan ^{2} \theta \csc ^{2} \theta-\tan ^{2} \theta = 1\\[4pt] \) 46. \( \sin ^{2} \theta \cot ^{2} \theta+\sin ^{2} \theta = 1\\[4pt] \) 47. \(3\sin^2 \theta + 4\cos^2 \theta =3+\cos^2\theta\\[4pt] \) 48. \(\sin^4 \;\theta ~-~ \cos^4 \;\theta ~=~ \sin^2 \;\theta ~-~ \cos^2 \;\theta\\[4pt] \) 49. \(\cos^2x-\tan^2x=2-\sin^2x-\sec^2x\\[4pt] \) 50. \(\cos^4 \;\theta ~-~ \sin^4 \;\theta ~=~ 1 ~-~ 2\;\sin^2 \;\theta\\[4pt] \) 51. \(\sec ^{4} \theta-\sec ^{2} \theta = \tan ^{4} \theta+\tan ^{2} \theta\\[4pt] \) 52. \(\csc ^{4} \theta-\csc ^{2} \theta = \cot ^{4} \theta+\cot ^{2} \theta \)

53. \(\cos \theta(\sec \theta-\cos \theta) = \sin ^{2} \theta\\[4pt] \) 54. \(\tan \theta(\cot \theta+\tan \theta) = \sec ^{2} \theta\\[4pt] \) 55. \(\tan \theta(\csc \theta+\cot \theta)-\sec \theta = 1\\[4pt] \) 56. \(\cot \theta(\sec \theta+\tan \theta)-\csc \theta = 1\\[4pt] \) 57. \(\cos \theta(\tan \theta+\cot \theta) = \csc \theta\\[4pt] \) 58. \(\sin \theta(\cot \theta+\tan \theta) = \sec \theta\\[4pt] \) 59. \(1 + \text{cot} (x) = \text{cos} (x) (\text{sec}(x) + \text{csc} (x)) \\[4pt] \) 60. \(\cos x(\tan x-\sec(-x))=\sin x-1\\[4pt] \) 61. \(\csc^2x(1-\sin^2x)=\cot^2x\\[4pt] \) 62. \( ( \sin \theta+\cos \theta)^{2}+(\sin \theta-\cos \theta)^{2} = 2\\[4pt] \) 63. \((\sin x+\cos x)^2=1+2 \sin x\cos x \)

Answers to odd exercises.

41. \( \cos \theta   \tan \theta  =  \dfrac{\cos \theta}{1} \cdot   \dfrac{\sin \theta}{\cos \theta} =\sin \theta \) 43. \( \cos x-\cos^3x = \cos x (1-\cos^2 x) = \cos x\sin ^x   \) 45.  \( \tan ^{2} \theta \csc ^{2} \theta-\tan ^{2} \theta \) =  \( \dfrac{\sin^2 \theta}{\cos^2 \theta} \cdot \dfrac{1}{\sin^2 \theta} - \dfrac{\sin^2 \theta}{\cos^2 \theta} \) =  \(  \dfrac{1-\sin^2 \theta}{\cos^2 \theta} \) =  \(  \dfrac{\cos^2 \theta}{\cos^2 \theta} \) =  \(  1 \) 47. \(  3\sin^2\theta + 4\cos^2\theta  = 3\sin ^2\theta +3\cos ^2\theta +\cos^2\theta  = 3\left ( \sin ^2\theta +\cos ^2\theta \right )+\cos^2\theta  = 3+\cos^2\theta \) 49. \(  \cos^2x-\tan^2x  = 1-\sin^2x-\left (\sec^2x -1 \right ) = 1-\sin^2x-\sec^2x +1 = 2-\sin^2x-\sec^2x   \) 51. \(\sec ^{4} \theta-\sec ^{2} \theta = \sec ^{2} \theta ( \sec ^{2} \theta-1) = (1+\tan^2 \theta ) (1+\tan^2 \theta  -1) = (1+\tan^2 \theta ) \tan^2 \theta  = \tan ^{4} \theta+\tan ^{2} \theta \)

53. \( \cos \theta(\sec \theta-\cos \theta) = \cos \theta \sec \theta-\cos^2 \theta = 1-  \cos^2 \theta  = \sin ^{2} \theta  \) 55. \(\tan \theta(\csc \theta+\cot \theta)-\sec \theta = \dfrac{\sin \theta}{\cos \theta} \cdot  \dfrac{1}{\sin \theta} + \dfrac{\sin \theta}{\cos \theta} \cdot  \dfrac{\cos \theta }{\sin \theta} - \dfrac{1}{\cos \theta}=  \dfrac{1}{\cos \theta} + 1 - \dfrac{1}{\cos \theta} = 1  \) 57. \(\cos \theta(\tan \theta+\cot \theta) = \dfrac{\cos \theta}{1} \cdot \dfrac{\sin \theta}{\cos \theta}+ \dfrac{\cos \theta}{1}  \cdot  \dfrac{\cos \theta }{\sin \theta} = \dfrac{ \sin^2 \theta}{\sin \theta} +\dfrac{ \cos^2 \theta}{\sin \theta}  =\dfrac{\sin^2 \theta+ \cos^2 \theta}{\sin \theta} = \dfrac{1}{\sin \theta} = \csc \theta  \) 59. \(  \text{cos} (x) (\text{sec}(x) + \text{csc} (x))  = \dfrac{cos \theta}{1} \cdot \dfrac{1}{cos \theta}+\dfrac{cos \theta}{1} \cdot  \dfrac{1}{\sin \theta} =1 + \cot \theta \) 61. \(\csc^2x(1-\sin^2x) =\csc^2x - 1 =\cot^2x + 1  - 1 = \cot^2 x  \) 63. \((\sin x+\cos x)^2 = \sin^2 + 2 \sin x \cos x + \cos^2 x =1+2 \sin x\cos x  \)

 

C: Verify (fractions)

Exercise \(\PageIndex{C}\)

\( \bigstar \) Verify the following identities.

71. \(\dfrac{\tan\;\theta}{\cot\;\theta} ~=~ \tan^2 \;\theta\\[4pt] \) 72. \(\dfrac{\csc\;\theta}{\sin\;\theta} ~=~ \csc^2 \;\theta\\[4pt] \) 73. \(\dfrac{\tan x}{\sec x}\sin (-x)=\cos ^2 x -1\\[4pt] \) 74. \(\dfrac{1 ~-~ 2\;\cos^2 \;\theta}{\sin\;\theta ~ \cos\;\theta} ~=~ \tan\;\theta ~-~ \cot\;\theta\  \) 75. \(\dfrac{1+\sin ^2x}{\cos ^2 x}=1+2\tan ^2x\\[4pt] \) 76. \(\dfrac{1 + \text{tan}^2(b)}{\text{tan}^2(b)} = \text{csc}^2(b) \)

77. \(\dfrac{\cos \theta}{\tan \theta}-\csc \theta = -\sin \theta\\[4pt] \) 78. \(\dfrac{\sin \theta}{\cot \theta}-\sec \theta = -\cos \theta \\[4pt]  \) 79. \(\text{tan}^{2} (t) = \dfrac{1}{\text{cos}^2(t)} - 1 \\[4pt] \) 80. \(\dfrac{(\sin \theta+\cos \theta)^{2}}{\cos \theta}-\sec \theta=2 \sin \theta\\[4pt] \) 81. \(\dfrac{(1 + \text{cos} A)(1 - \text{cos} A)}{\text{sin} A} = \text{sin} A  \)

Answers to odd exercises.

71. \(\dfrac{\tan \theta}{\cot \theta} = \tan \theta  \cdot \dfrac{1}{\cot \theta} = \tan \theta  \cdot  \tan \theta = \tan^2  \theta  \) 73. \(  \dfrac{\tan x}{\sec x}\sin (-x) =\dfrac{\sin x}{\cos x} \cdot \dfrac{\cos x}{1} \cdot \sin x = -sin^2 x = - (1- \cos^2 x)   =\cos ^2 x -1  \) 75. \(  \dfrac{1+\sin ^2 x}{\cos ^2 x} = \dfrac{1}{\cos ^2 x}+\dfrac{\sin ^2x}{\cos ^2 x} = \sec ^2x+\tan ^2x = \tan ^2x+1+\tan ^2x = 1+2\tan ^2x \) 77. \(\dfrac{\cos \theta}{\tan \theta}-\csc \theta =  \dfrac{\cos \theta}{1}  \cdot  \dfrac{\cos \theta}{\sin \theta} - \dfrac{1}{\sin \theta}   =  \dfrac{\cos^2  \theta - 1}{\sin \theta}  = \dfrac{\sin^2 \theta}{\sin \theta} = -\sin \theta \) 79. \(\text{tan}^{2} (t) = \dfrac{1}{\text{cos}^2(t)} - 1  = \dfrac{ 1}{ \cos^2 t} - \dfrac{ \cos^2 t}{ \cos^2 t} = \dfrac{ \cos^2 t - 1}{ \cos^2 t} = \dfrac{ -(1-\cos^2 t)}{ \cos^2 t} =\dfrac{ \sin^2 t }{ \cos^2 t} = \tan^2 t  \) 81. \(\dfrac{(1 + \text{cos}(A))(1 - \text{cos} (A))}{\text{sin} (A)} = \dfrac{1 - \cos^2 (A))}{\sin (A)} = \dfrac{ \sin^2 (A))}{\sin (A)} = \sin (A)  \)

\( \bigstar \) Verify the following identities.

82. \(\dfrac{\csc \theta}{\cos \theta}-\dfrac{\cos \theta}{\csc \theta} = \dfrac{\cot ^{2} \theta+\sin ^{2} \theta}{\cot \theta}\\[4pt] \)  83. \(\dfrac{\sec \theta}{\csc \theta} + \dfrac{\sin \theta}{\cos \theta} = 2 \tan \theta\\[4pt] \) 84. \(\dfrac{\text{csc}^2 (\alpha) - 1}{\text{csc}^2 (\alpha) - \text{csc} (\alpha)} = 1 + \text{sin} (\alpha) \\[4pt] \) 85. \(\dfrac{\sin \theta \tan \theta+\sin \theta}{\tan \theta+\tan ^{2} \theta} = \cos \theta\\[4pt] \)

86. \(\dfrac{\cos \theta \cot \theta+\cos \theta}{\cot \theta+\cot ^{2} \theta} = \sin \theta\\[4pt] \) 87. \(\dfrac{\sin^2 \;\theta}{1 ~-~ \sin^2 \;\theta} ~=~ \tan^2 \;\theta\\[4pt] \) 88. \(\dfrac{1 ~-~ \tan^2 \;\theta}{1 ~-~ \cot^2 \;\theta} ~=~ 1 ~-~ \sec^2 \;\theta\\[4pt] \) 89. \(\dfrac{\sec ^{2} \theta}{1+\cot ^{2} \theta} = \tan ^{2} \theta\\[4pt] \)

90. \(\dfrac{\cos ^2 \theta -\sin ^2 \theta }{1-\tan ^2 \theta }= 1 - \sin ^2 \theta\\[4pt] \) 91. \(\dfrac{\csc ^{2} \theta}{1+\tan ^{2} \theta} = \cot ^{2} \theta\\[4pt] \) 92. \(\sin\;\theta ~=~ \pm\,\dfrac{\tan\;\theta}{\sqrt{1 ~+~ \tan^2 \;\theta}}\qquad \\[4pt] \) 93. \(\dfrac{\sec \theta +\tan \theta }{\sin \theta(\cot \theta+\cos \theta) }=\sec ^2 \theta\\[4pt] \)

Answers to odd exercises.

83. \( \dfrac{\sec \theta}{\csc \theta} + \dfrac{\sin \theta}{\cos \theta} = \dfrac{\sec \theta}{1} \cdot \dfrac{1}{\csc \theta} + \dfrac{\sin \theta}{\cos \theta} =\dfrac{1}{\cos  \theta}\cdot \dfrac{\sin \theta}{1} + \dfrac{\sin \theta}{\cos \theta} = 2 \dfrac{\sin \theta}{\cos \theta}= 2 \tan \theta \\[2pt] \) 85. \(\dfrac{\sin \theta \tan \theta+\sin \theta}{\tan \theta+\tan ^{2} \theta} = \dfrac{\sin \theta ( \tan \theta+1)}{\tan \theta(1 + \tan \theta) } = \sin \theta \cdot \cot \theta =\sin \theta \cdot \dfrac{\cos \theta}{\sin \theta} = \cos \theta  \) 87. \(\dfrac{\sin^2 \;\theta}{1 ~-~ \sin^2 \;\theta} = \dfrac{\sin^2 \theta}{ \cos^2 \theta} = \left(  \dfrac{\sin \theta}{\cos \theta}  \right)^2  = \tan^2 \;\theta  \) 89. \(\dfrac{\sec ^{2} \theta}{1+\cot ^{2} \theta} = \dfrac{\sec ^{2} \theta}{\csc ^{2} \theta} = \dfrac{1}{\cos^2 \theta} \cdot \dfrac{\sin^2 \theta}{1}   =  \tan ^{2} \theta  \) 91. \(\dfrac{\csc ^{2} \theta}{1+\tan ^{2} \theta} =  \dfrac{\csc ^{2} \theta}{\sec ^{2} \theta} = \dfrac{1}{\sin^2 \theta} \cdot \dfrac{\cos^2 \theta}{1}    = \cot ^{2} \theta  \)

93. \( \dfrac{\sec \theta +\tan \theta }{\sin \theta(\cot \theta+\cos \theta) } = \dfrac{    \dfrac{1}{\cos \theta} +  \dfrac{\sin \theta}{\cos \theta}   }{\sin \theta \left(  \dfrac{\cos \theta}{\sin \theta} +\cos \theta  \right) } = \dfrac{    \dfrac{1}{\cos \theta} (1+  \sin \theta)   }{\cos \theta \sin \theta \left(  \dfrac{1}{\sin \theta} + 1 \right) } = \dfrac{    \dfrac{1}{\cos \theta} (1+  \sin \theta)   }{\cos \theta \left(  1 + \sin \theta \right) }   =\dfrac{1}{\cos^2 \theta} = \sec ^2 \theta  \)

D: Verify (more fractions)

Exercise \(\PageIndex{D}\)

\( \bigstar \) Verify the following identities.

101. \(\dfrac{1+\sin x}{\cos x}=\dfrac{\cos x}{1+\sin (-x)}\\[4pt] \) 102. \(\dfrac{\cos^2 \;\theta}{1 ~+~ \sin\;\theta} ~=~ 1 ~-~ \sin\;\theta\\[4pt] \) 103. \(\dfrac{1 + \text{cos} (u)}{\text{sin} (u)} = \dfrac{\text{sin} (u)}{1 - \text{cos}(u)} \\[4pt] \) 104. \(\dfrac{\sin ^{2} \left(\theta \right)}{1+\cos \left(\theta \right)} =1-\cos \left(\theta \right) \\[4pt] \) 105. \(\dfrac{1 ~-~ \tan\;\theta}{1 ~+~ \tan\;\theta} ~=~ \dfrac{\cot\;\theta ~-~ 1}{\cot\;\theta ~+~ 1}\\[4pt] \) 106. \(\dfrac{\text{sin}^4 (\gamma) - \text{cos}^4 (\gamma)}{\text{sin} (\gamma) - \text{cos} (\gamma)} = \text{sin} (\gamma) + \text{cos} (\gamma) \\[4pt] \) 107. \(\dfrac{\text{csc}^2 (x) - \text{sin}^2 (x)}{\text{csc} (x) + \text{sin} (x)} = \text{cos} (x) \text{cot} (x) \\[4pt] \)

108. \(\dfrac{\sec \theta-\cos \theta}{\sec \theta+\cos \theta} = \dfrac{\sin ^{2} \theta}{1+\cos ^{2} \theta}\\[4pt] \) 109. \(\dfrac{\sec \theta+\tan \theta}{\cot \theta+\cos \theta} = \tan \theta \sec \theta\\[4pt] \) 110. \(\dfrac{\tan\;\theta ~+~ \tan\;\phi}{\cot\;\theta ~+~ \cot\;\phi} ~=~ \tan\;\theta ~ \tan\;\phi\\[4pt] \) 111. \(\dfrac{\sec (-x)}{\tan x+\cot x}=-\sin (-x)\\[4pt] \) 112. \(\dfrac{\sec \theta+\csc \theta}{\tan \theta+\cot \theta} = \sin \theta+\cos \theta\\[4pt] \)  113. \(\dfrac{\tan \theta-\cot \theta}{\tan \theta+\cot \theta} = \sin ^{2} \theta-\cos ^{2} \theta\\[4pt] \) 114. \(\dfrac{\text{sin} (\theta) - \text{cos} (\theta)}{\text{sec}(\theta) - \text{csc} (\theta)} = \text{sin} (\theta) \text{cos} (\theta) \\[4pt] \)

Answers to odd exercises.

101. \(=\dfrac{\cos x}{1-\sin (x)} \times \dfrac{1+ \sin x}{1+ \sin x} = \dfrac{\cos x (1 + \sin x)}{1-\sin^2 x} = \dfrac{\cos x (1 + \sin x)}{\cos^2 x} = \dfrac{1+\sin x}{\cos x}  \) 103. \(\dfrac{1 + \text{cos} (u)}{\text{sin} (u)} =  \dfrac{1 + \text{cos} (u)}{\text{sin} (u)} \cdot \dfrac{\sin u}{\sin u} =  \dfrac{\sin u (1 + \cos u) }{\sin^2 u} =  \dfrac{\sin u (1 + \cos u) }{1-\cos^2 u} =  \dfrac{\sin u (1 + \cos u) }{(1-\cos u)(1+\cos u)} \\ \) \( \quad =  \dfrac{\text{sin} (u)}{1 - \text{cos}(u)} \) 105. \(\dfrac{1 ~-~ \tan\;\theta}{1 ~+~ \tan\;\theta} =  \dfrac{(1 ~-~ \tan\;\theta)}{(1 ~+~ \tan\;\theta)}  \cdot \dfrac{\cot  \theta}{\cot \theta}  =   \dfrac{\cot\;\theta ~-~ 1}{\cot\;\theta ~+~ 1} \) 107. \(\dfrac{\text{csc}^2 (x) - \text{sin}^2 (x)}{\text{csc} (x) + \text{sin} (x)} =  \dfrac{( \csc x - \sin x)(\csc x + \sin x)}{(\csc x + \sin x)}\ = \csc x - \sin x = \dfrac{1}{\sin x} -  \dfrac{\sin^2 x}{\sin x} = \dfrac{1 - \sin^2 x}{\sin x} \\ \) \( \quad  = \dfrac{ \cos^2 x}{\sin x} = \cos x \cdot \dfrac{\cos x}{ \sin x} = \text{cos} (x) \text{cot} (x) \ \) 109. \(\dfrac{\sec \theta+\tan \theta}{\cot \theta+\cos \theta} =  \dfrac{\dfrac{1}{\cos \theta}+\dfrac{\sin \theta}{\cos \theta}}{\dfrac{\cos \theta}{\sin \theta}+\dfrac{\cos \theta \sin \theta}{\sin \theta}} =  \dfrac{\dfrac{1+\sin \theta}{\cos \theta}}{\dfrac{\cos \theta \left( 1+ \sin \theta \right)}{\sin \theta} } =  \dfrac{1+\sin \theta}{\cos \theta} \cdot \dfrac{\sin \theta}{\cos \theta \left( 1+ \sin \theta \right)} = \dfrac{\sin \theta }{\cos^2 \theta } \\ \) \( \quad  = \dfrac{\sin \theta }{\cos \theta } \cdot \dfrac{1}{\cos \theta}= \tan \theta \sec \theta \) 111. \(\dfrac{\sec (-x)}{\tan x+\cot x} =  \dfrac{\sec (x)}{\dfrac{\sin x}{\cos x}+\dfrac{\cos x}{\sin x}} =  \dfrac{\dfrac{1}{\cos x}}{\dfrac{\sin^2 x + \cos^2 x}{\cos x \sin x}} =  \dfrac{\dfrac{1}{\cos x}}{\dfrac{1}{\cos x \sin x}} =\dfrac{1}{\cos x} \cdot \dfrac{\cos x \sin x}{1}  = \sin x  =  -\sin (-x)\\[4pt] \) 113. \(\dfrac{\tan \theta-\cot \theta}{\tan \theta+\cot \theta} = \dfrac{\dfrac{\sin \theta}{\cos \theta}-\dfrac{\cos \theta}{\sin \theta}}{\dfrac{\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{\sin \theta}} =\dfrac{\dfrac{\sin^2 \theta - \cos^2 \theta}{\sin \theta\cos \theta}}{\dfrac{\sin^2 \theta + \cos^2 \theta}{\sin \theta\cos \theta}}= \dfrac{\dfrac{\sin^2 \theta - \cos^2 \theta}{\sin \theta\cos \theta}}{\dfrac{1}{\sin \theta\cos \theta}} = \dfrac{\sin^2 \theta - \cos^2 \theta}{\sin \theta\cos \theta} \cdot \dfrac{\sin \theta\cos \theta}{1} \\ \) \( \quad  =  \sin ^{2} \theta-\cos ^{2} \theta \)

 \( \bigstar \) Verify the following identities.

115. \(\dfrac{\sin \theta}{1+\sin \theta}-\dfrac{\sin \theta}{1-\sin \theta} = -2 \tan ^{2} \theta\\[4pt] \) 116. \(\dfrac{\cos \theta}{1+\cos \theta}-\dfrac{\cos \theta}{1-\cos \theta} = -2 \cot ^{2} \theta\\[4pt] \) 117. \(\dfrac{1}{1+\cos x}-\dfrac{1}{1-\cos (-x)}=-2\cot x\csc x\\[4pt] \) 118. \(\dfrac{\cot \theta}{1-\csc \theta}-\dfrac{\cot \theta}{1+\csc \theta} = 2 \sec \theta\\[4pt] \)  119. \(\dfrac{\tan \theta}{1+\sec \theta}-\dfrac{\tan \theta}{1-\sec \theta} = 2 \csc \theta\\[4pt] \) 120. \(\dfrac{1-\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{1-\sin \theta} = 2 \sec \theta\\[4pt] \)

121. \(\dfrac{\cos \theta}{1+\sin \theta}+\dfrac{1+\sin \theta}{\cos \theta} = 2 \sec \theta\\[4pt] \) 122. \(1-\dfrac{\cos ^{2} \theta}{1+\sin \theta} = \sin \theta\\[4pt] \) 123. \(1-\dfrac{\sin ^{2} \theta}{1+\cos \theta} = \cos \theta\\[4pt] \) 124. \(2 \text{sec}^2 (t) = \dfrac{1 - \text{sin}(t)}{\text{cos}^2 (t)} + \dfrac{1}{1 -  \text{sin} (t)} \\[4pt] \) 125. \(\left (\dfrac{\sec ^2(-x)-\tan ^2x}{\tan x} \right )\left (\dfrac{2+2\tan x}{2+2\cot x} \right )-2\sin ^2x \\ \)      \( \qquad =\cos^2 x - \sin^2 x \\[4pt] \)

Answers to odd exercises.

115. \(\dfrac{\sin \theta}{1+\sin \theta}-\dfrac{\sin \theta}{1-\sin \theta} =  \dfrac{\sin \theta}{(1+\sin \theta)} \cdot \dfrac{(1-\sin \theta)}{(1-\sin \theta)} - \dfrac{\sin \theta}{(1-\sin \theta)} \cdot \dfrac{(1+\sin \theta)}{(1+\sin \theta)} \\ \) \( \qquad  = \dfrac{(\sin \theta -\sin^2 \theta) - (\sin \theta +\sin^2 \theta) }{1 - \sin^2 \theta} = \dfrac{-2 \sin^2 \theta}{\cos^2 \theta} =    -2 \tan ^{2} \theta \) 117. \(\dfrac{1}{1+\cos x}-\dfrac{1}{1-\cos (-x)} =  \dfrac{1}{(1+\cos x)} \dfrac{(1 - \cos x)}{(1 - \cos x)}  -\dfrac{1}{(1-\cos x)}\dfrac{(1 + \cos x)}{(1 + \cos x)}  = \dfrac{(1 - \cos x) - (1 + \cos x)}{1-\cos^2 x} \\ \) \( \qquad  = \dfrac{ -2 \cos x }{ \sin^2 x} =   \dfrac{ -2 \cos x }{ \sin x} \cdot \dfrac{ 1}{ \sin x}  =-2\cot x\csc x \) 119. \(\dfrac{\tan \theta}{1+\sec \theta}-\dfrac{\tan \theta}{1-\sec \theta} = \dfrac{\tan \theta}{(1+\sec \theta)}\dfrac{(1-\sec \theta)}{(1-\sec \theta)} -\dfrac{\tan \theta}{(1-\sec \theta)} \dfrac{(1+\sec \theta)}{(1+\sec \theta)} = \dfrac{-2 \tan \theta \sec \theta }{1-\sec^2 \theta } = \dfrac{-2 \tan \theta \sec \theta }{-\tan^2 \theta } \\ \) \( \qquad = \dfrac{2  \sec \theta }{\tan\theta } =   \dfrac{2  \sec \theta }{1 } \cdot \dfrac{\cos \theta }{\sin\theta } =\dfrac{2}{\sin\theta} =  2 \csc \theta \) 121. \(\dfrac{\cos \theta}{1+\sin \theta}+\dfrac{1+\sin \theta}{\cos \theta} = \dfrac{\cos \theta}{(1+\sin \theta)}\dfrac{\cos \theta}{\cos \theta}+\dfrac{(1+\sin \theta)}{\cos \theta} \dfrac{(1+\sin \theta)}{(1+\sin \theta)} = \dfrac{\cos^2 \theta + 1 + 2 \sin \theta + \sin^2 \theta}{\cos \theta (1+\sin \theta)}  \\ \) \( \qquad  = \dfrac{2 + 2 \sin \theta }{\cos \theta (1+\sin \theta)} = \dfrac{2(1+ \sin \theta )}{\cos \theta (1+\sin \theta)}= \dfrac{2}{\cos \theta}  =  2 \sec \theta \) 123. \(1-\dfrac{\sin ^{2} \theta}{1+\cos \theta} =1-\dfrac{1-\cos^{2} \theta}{1+\cos \theta} =1-\dfrac{(1-\cos \theta)(1+\cos \theta)}{1+\cos \theta} = 1 - (1-\cos \theta)=  \cos \theta \) 125. \(\left (\dfrac{\sec ^2(-x)-\tan ^2x}{\tan x} \right )\left (\dfrac{2+2\tan x}{2+2\cot x} \right )-2\sin ^2x =  \left (\dfrac{1+\tan ^2x -\tan ^2x}{\tan x} \right )  \dfrac{2 \left( 1+    \dfrac{\sin x}{\cos x}     \right) }{  2 \left( 1+    \dfrac{\cos x}{\sin x}     \right)  } -2\sin ^2x \\ \) \( \qquad  =  \dfrac{1}{\tan x} \cdot  \dfrac{ \dfrac{\cos x +\sin x}{\cos x}}{\dfrac{\sin x+\cos x}{\sin x} }  - \sin^2 x=    \dfrac{\cos x}{\sin x} \cdot \dfrac{\cos x +\sin x}{\cos x} \cdot \dfrac{\sin x}{\sin x+\cos x}  -2\sin ^2x = 1 -2\sin ^2x \\ \) \( \qquad  = \cos^2 x + \sin^2 x  -2\sin ^2x = \cos^2 x - \sin^2 x \)

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