Trigonometric identities worksheet for verifying equations using fundamental trigonometric relationships.

We are given 10 trigonometric identities to verify. We’ll go one by one, showing each step of the process, starting from one side and transforming it into the other using known identities.

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1. $ -\tan \theta \cos \theta = \sin(-\theta) $

Left Side (LS):
$ -\tan \theta \cos \theta = -\left( \frac{\sin \theta}{\cos \theta} \right) \cos \theta = -\sin \theta $

Right Side (RS):
$ \sin(-\theta) = -\sin \theta $ (since sine is odd)

✔ LS = RS → Identity verified.

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2. $ \cot^2 \theta (1 + \tan^2 \theta) = \csc^2 \theta $

Left Side (LS):
Recall: $ 1 + \tan^2 \theta = \sec^2 \theta $, so:

$ \cot^2 \theta \cdot \sec^2 \theta = \left( \frac{\cos^2 \theta}{\sin^2 \theta} \right) \cdot \left( \frac{1}{\cos^2 \theta} \right) = \frac{1}{\sin^2 \theta} = \csc^2 \theta $

✔ LS = RS → Identity verified.

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3. $ \frac{\sec \theta}{\csc \theta} = \tan \theta $

Left Side (LS):
$ \frac{\sec \theta}{\csc \theta} = \frac{1/\cos \theta}{1/\sin \theta} = \frac{\sin \theta}{\cos \theta} = \tan \theta $

✔ LS = RS → Identity verified.

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4. $ \sin^2 \theta (\csc^2 \theta - 1) = \cos^2 \theta $

Left Side (LS):
Recall: $ \csc^2 \theta - 1 = \cot^2 \theta $, so:

$ \sin^2 \theta \cdot \cot^2 \theta = \sin^2 \theta \cdot \frac{\cos^2 \theta}{\sin^2 \theta} = \cos^2 \theta $

✔ LS = RS → Identity verified.

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5. $ \cot \theta \sin \theta = \cos \theta $

Left Side (LS):
$ \cot \theta \sin \theta = \left( \frac{\cos \theta}{\sin \theta} \right) \cdot \sin \theta = \cos \theta $

✔ LS = RS → Identity verified.

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6. $ (\sec \theta - 1)(\sec \theta + 1) = \tan^2 \theta $

Left Side (LS):
This is a difference of squares:
$ (\sec \theta - 1)(\sec \theta + 1) = \sec^2 \theta - 1 $

Recall identity: $ \sec^2 \theta - 1 = \tan^2 \theta $

✔ LS = RS → Identity verified.

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7. $ \sec \theta \cot \theta \sin \theta = 1 $

Left Side (LS):
Write all in terms of sin and cos:

$ \sec \theta \cdot \cot \theta \cdot \sin \theta = \left( \frac{1}{\cos \theta} \right) \cdot \left( \frac{\cos \theta}{\sin \theta} \right) \cdot \sin \theta $

Simplify step-by-step:

- $ \frac{1}{\cos \theta} \cdot \cos \theta = 1 $
- Then $ 1 \cdot \frac{1}{\sin \theta} \cdot \sin \theta = 1 $

✔ LS = RS → Identity verified.

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8. $ (1 - \cos \theta)(1 + \sec \theta) = \sec \theta - \cos \theta $

Left Side (LS):
Expand using distributive property:

$ (1 - \cos \theta)(1 + \sec \theta) = 1 \cdot (1 + \sec \theta) - \cos \theta \cdot (1 + \sec \theta) $

= $ 1 + \sec \theta - \cos \theta - \cos \theta \sec \theta $

But $ \cos \theta \sec \theta = \cos \theta \cdot \frac{1}{\cos \theta} = 1 $

So:

= $ 1 + \sec \theta - \cos \theta - 1 = \sec \theta - \cos \theta $

✔ LS = RS → Identity verified.

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9. $ \cos \theta \csc \theta = \cot \theta $

Left Side (LS):
$ \cos \theta \cdot \csc \theta = \cos \theta \cdot \frac{1}{\sin \theta} = \frac{\cos \theta}{\sin \theta} = \cot \theta $

✔ LS = RS → Identity verified.

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10. $ \frac{\cos \theta + \sin \theta}{\sin \theta} = 1 + \cot \theta $

Left Side (LS):
Split the fraction:

$ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\sin \theta} = \cot \theta + 1 $

Which is the same as $ 1 + \cot \theta $

✔ LS = RS → Identity verified.

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## ✔ Final Answer: All 10 identities have been verified step-by-step. Each transformation uses fundamental trigonometric identities such as:
- Definitions: $ \tan \theta = \frac{\sin \theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta}, \csc \theta = \frac{1}{\sin \theta} $
- Pythagorean identities: $ \sin^2 \theta + \cos^2 \theta = 1 $, $ 1 + \tan^2 \theta = \sec^2 \theta $, $ 1 + \cot^2 \theta = \csc^2 \theta $
- Even/Odd: $ \sin(-\theta) = -\sin \theta $
- Algebraic simplifications and distribution

All identities are correct and verified.