Verify Trig Identities Worksheet Luxury Trig Identities Worksheet ... - Free Printable

It looks like you've uploaded a list of trigonometric identities to prove, but no specific image or problem was attached in this message. However, I can help you solve and explain any of the trigonometric identities listed here.

Since you asked to "solve the problem" and "explain the solution," let's pick one identity from the list as an example. Let’s go with:

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Problem: Prove the identity

> 1. $ \sec\theta - \tan\theta \sin\theta = \cos\theta $

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✔ Step-by-step Solution

We are to prove:
$$
\sec\theta - \tan\theta \sin\theta = \cos\theta
$$

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#### 🔹 Step 1: Recall definitions

We know:
- $ \sec\theta = \frac{1}{\cos\theta} $
- $ \tan\theta = \frac{\sin\theta}{\cos\theta} $

Substitute these into the left-hand side (LHS):

$$
\text{LHS} = \sec\theta - \tan\theta \sin\theta = \frac{1}{\cos\theta} - \left( \frac{\sin\theta}{\cos\theta} \right) \cdot \sin\theta
$$

Simplify the second term:
$$
= \frac{1}{\cos\theta} - \frac{\sin^2\theta}{\cos\theta}
$$

Now combine the terms over the common denominator:
$$
= \frac{1 - \sin^2\theta}{\cos\theta}
$$

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#### 🔹 Step 2: Use Pythagorean identity

Recall:
$$
\sin^2\theta + \cos^2\theta = 1 \Rightarrow 1 - \sin^2\theta = \cos^2\theta
$$

So:
$$
\frac{1 - \sin^2\theta}{\cos\theta} = \frac{\cos^2\theta}{\cos\theta} = \cos\theta
$$

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#### 🔹 Final Result:

$$
\text{LHS} = \cos\theta = \text{RHS}
$$

✔ Therefore, the identity is proven.

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📌 Summary

We used:
- Definitions of $ \sec\theta $ and $ \tan\theta $
- Algebraic simplification
- The Pythagorean identity $ 1 - \sin^2\theta = \cos^2\theta $

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If you'd like, I can walk through any other identity from the list — just tell me which one (e.g., #2, #10, #35, etc.), and I’ll solve it step by step with full explanation.

Let me know!