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Let's tackle the trigonometric identity proofs from the image step by step. The question asks to prove the following identities:

### a) \( \sec t - \sin t \tan t = \cos t \)

**Proof:**
Start with the left-hand side (LHS):
\[
\sec t - \sin t \tan t
\]
- Recall that \( \sec t = \frac{1}{\cos t} \) and \( \tan t = \frac{\sin t}{\cos t} \), so we substitute these:
\[
\frac{1}{\cos t} - \sin t \cdot \frac{\sin t}{\cos t}
\]
- Combine the terms under a common denominator:
\[
\frac{1 - \sin^2 t}{\cos t}
\]
- Now use the Pythagorean identity \( 1 - \sin^2 t = \cos^2 t \):
\[
\frac{\cos^2 t}{\cos t}
\]
- Simplify:
\[
\cos t
\]
Thus, the left-hand side is equal to the right-hand side (RHS), so the identity is proven.

### b) \( (1 + \sin z)(1 - \sin z) = \frac{1}{\sec^2 z} \)

**Proof:**
Start with the left-hand side:
\[
(1 + \sin z)(1 - \sin z)
\]
- This is a difference of squares:
\[
1^2 - \sin^2 z = 1 - \sin^2 z
\]
- Use the Pythagorean identity \( \sin^2 z + \cos^2 z = 1 \), which gives:
\[
1 - \sin^2 z = \cos^2 z
\]
Thus, the left-hand side becomes:
\[
\cos^2 z
\]
Now, simplify the right-hand side:
\[
\frac{1}{\sec^2 z} = \frac{1}{\frac{1}{\cos^2 z}} = \cos^2 z
\]
Since both sides are equal, the identity is proven.

### c) \( (\sec t - 1)(\sec t + 1) = \tan^2 t \)

**Proof:**
Start with the left-hand side:
\[
(\sec t - 1)(\sec t + 1)
\]
- This is a difference of squares:
\[
\sec^2 t - 1
\]
- Use the identity \( \sec^2 t - 1 = \tan^2 t \):
\[
\tan^2 t
\]
So the identity is proven.

### d) \( \frac{\sec^2 t - 1}{\sec^2 t} = \sin^2 t \)

**Proof:**
Start with the left-hand side:
\[
\frac{\sec^2 t - 1}{\sec^2 t}
\]
- Using the identity \( \sec^2 t - 1 = \tan^2 t \), substitute this into the equation:
\[
\frac{\tan^2 t}{\sec^2 t}
\]
- Now use the identity \( \tan^2 t = \frac{\sin^2 t}{\cos^2 t} \) and \( \sec^2 t = \frac{1}{\cos^2 t} \), so:
\[
\frac{\frac{\sin^2 t}{\cos^2 t}}{\frac{1}{\cos^2 t}} = \sin^2 t
\]
Thus, the left-hand side equals the right-hand side, and the identity is proven.

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Would you like more details on any specific step?

Here are 5 related questions:
1. How do you prove other trigonometric identities?
2. What are the key Pythagorean identities in trigonometry?
3. How can difference of squares be applied to trigonometric functions?
4. What is the relationship between \( \sec t \) and \( \cos t \)?
5. How does the identity \( \tan^2 t = \sec^2 t - 1 \) relate to basic trigonometric functions?

**Tip:** When working with trigonometric identities, remember that substitution and using Pythagorean identities are powerful tools for simplifying expressions.

Prove the following trigonometric identities:
 a) sec t - sin t tan t = cos t
 b) (1 + sin z)(1 - sin z) = 1 / sec^2 z
 c) (sec t - 1)(sec t + 1) = tan^2 t
 d) (sec^2 t - 1) / sec^2 t = sin^2 t

Explore step-by-step solutions for proving trigonometric identities involving sec t, sin t, cos t, and tan t. Learn how to simplify using key trigonometric formulas such as sec t = 1 / cos t and the Pythagorean identity. Suitable for students in Grades 10-12.

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