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# 7.1E: Verifying Trigonometric Identities

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### 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. $$\left (\dfrac{\tan x}{\csc ^2 x}+\dfrac{\tan x}{\sec ^2 x} \right )\left (\dfrac{1+\tan x}{1+\cot x} \right )-\dfrac{1}{\cos ^2 x}\\[4pt]$$

1. sec($$t \$$)     3. $$\sin x$$     5. $$\cos x\$$    7. $$-1$$     9. tan($$t$$)     11. tan($$t$$)
13. $$-1$$      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. $$\dfrac{1}{1-\cos x}-\dfrac{\cos x}{1+\cos x}$$;  $$f( \csc x) \\[4pt]$$

25. $$\pm \sqrt{\dfrac{1}{\cot ^2 x}+1}$$     27. $$\dfrac{\pm \sqrt{1-\sin ^2 x}}{\sin x}$$     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. $$(\sin x+\cos x)^2=1+2 \sin x\cos x$$
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$$
71. $$\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. $$\dfrac{\sec \theta +\tan \theta }{\sin \theta(\cot \theta+\cos \theta) }=\sec ^2 \theta\\[4pt]$$
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. $$\dfrac{\text{sin} (\theta) - \text{cos} (\theta)}{\text{sec}(\theta) - \text{csc} (\theta)} = \text{sin} (\theta) \text{cos} (\theta) \\[4pt]$$

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. $$\left (\dfrac{\sec ^2(-x)-\tan ^2x}{\tan x} \right )\left (\dfrac{2+2\tan x}{2+2\cot x} \right )-2\sin ^2x \\$$      $$\qquad =\cos^2 x - \sin^2 x \\[4pt]$$
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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