The image shows a set of trigonometric identities that need to be verified. Let's go through each one step by step:

(a) $(1 + \sin t)(1 - \sin t) = \frac{1}{\sec^2 t}$

Verification:

1. Expand the left-hand side: $(1 + \sin t)(1 - \sin t) = 1^2 - (\sin t)^2 = 1 - \sin^2 t$
2. Use the Pythagorean identity: $\sin^2 t + \cos^2 t = 1$, hence: $1 - \sin^2 t = \cos^2 t$
3. Now, the equation becomes: $\cos^2 t = \frac{1}{\sec^2 t}$ which is true, since $\sec t = \frac{1}{\cos t}$.

Thus, the identity is verified.

(b) $(\sec t - 1)(\sec t + 1) = \tan^2 t$

Verification:

1. Expand the left-hand side: $(\sec t - 1)(\sec t + 1) = \sec^2 t - 1$
2. Use the identity $\sec^2 t = 1 + \tan^2 t$: $\sec^2 t - 1 = \tan^2 t$ which matches the right-hand side.

Thus, the identity is verified.

(c) $\sec t - \sin t \tan t = \cos t$

Verification:

1. Use the identity for $\tan t = \frac{\sin t}{\cos t}$: $\sec t - \sin t \tan t = \frac{1}{\cos t} - \sin t \cdot \frac{\sin t}{\cos t} = \frac{1 - \sin^2 t}{\cos t}$
2. Recognize that $1 - \sin^2 t = \cos^2 t$: $\frac{\cos^2 t}{\cos t} = \cos t$

Thus, the identity is verified.

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

Verification:

1. Simplify the left-hand side: $\frac{\sec^2 t - 1}{\sec^2 t} = 1 - \frac{1}{\sec^2 t}$
2. Use the identity $\sec^2 t = 1 + \tan^2 t$ and $\sin^2 t = \frac{\tan^2 t}{\sec^2 t}$: $1 - \cos^2 t = \sin^2 t$ which is true.

Thus, the identity is verified.

(e) $\cos t (\tan t + \cot t) = \csc t$

Verification:

1. Express $\tan t$ and $\cot t$ in terms of $\sin t$ and $\cos t$: $\tan t = \frac{\sin t}{\cos t}, \quad \cot t = \frac{\cos t}{\sin t}$
2. The left-hand side becomes: $\cos t \left( \frac{\sin t}{\cos t} + \frac{\cos t}{\sin t} \right) = \sin t + \frac{\cos^2 t}{\sin t}$ Simplifying further: $\sin t \cdot \frac{1}{\sin t} = \csc t$

Thus, the identity is verified.

Would you like more details on any of these steps or have any questions?

Relative Questions

1. How does the identity $\sin^2 t + \cos^2 t = 1$ help in proving other trigonometric identities?
2. Can you provide a geometric interpretation of the trigonometric identity $\sec^2 t - 1 = \tan^2 t$?
3. Why is it important to express trigonometric identities in different forms?
4. What are some common mistakes to avoid when verifying trigonometric identities?
5. How does understanding trigonometric identities help in solving calculus problems?

Tip

When verifying trigonometric identities, always start by expressing all functions in terms of sine and cosine, as this can simplify the verification process.