2) Prove the following identities a) \( \left(1+\tan ^{2} x\right) \cos ^{2} x=1 \) b) \( \sin ^{2} x+\frac{\sin x \cos x}{\tan x}=1 \) c) \( \frac{1}{1-\sin x}-\frac{1}{1+\sin x}=\frac{2}{\text { c }} \) d) \( \frac{\tan ^{2} \alpha}{1+\tan ^{2} \alpha}=\sin ^{2} \alpha \)

We will prove each identity step by step. ───────────────────────────── (a) Prove: (1 + tan²x) cos²x = 1 Step 1. Recall that tan x = sin x/cos x. But more directly, remember the Pythagorean identity:   1 + tan²x = sec²x. Step 2. Since sec x is the reciprocal of cos x, we have sec²x = 1/cos²x. Step 3. Substitute this into the left‐hand side:   (1 + tan²x) cos²x = sec²x cos²x = (1/cos²x) cos²x = 1. Thus, the identity is proved. ───────────────────────────── (b) Prove: sin²x + (sin x cos x)⁄tan x = 1 Step 1. Write tan x as sin x/cos x. Then the term (sin x cos x)/tan x becomes:   (sin x cos x)/(sin x/cos x) Step 2. Simplify the fraction:   (sin x cos x) × (cos x/sin x) = cos²x    (sin x cancels). Step 3. Now the original expression becomes:   sin²x + cos²x. Step 4. Using the Pythagorean identity:   sin²x + cos²x = 1. Thus, the given identity is established. ───────────────────────────── (c) Prove: 1/(1 − sin x) − 1/(1 + sin x) = (2 sin x)/(cos²x) Step 1. Find a common denominator: The denominators (1 − sin x) and (1 + sin x) multiply to give:   (1 − sin x)(1 + sin x) = 1 − sin²x. Step 2. Write the difference as:   [ (1 + sin x) − (1 − sin x) ]⁄(1 − sin²x). Step 3. Simplify the numerator:   (1 + sin x) − (1 − sin x) = 2 sin x. Step 4. So the expression becomes:   (2 sin x)/(1 − sin²x). Step 5. But recall that 1 − sin²x = cos²x. Therefore:   (2 sin x)/(cos²x). Thus, the identity is proved. ───────────────────────────── (d) Prove: (tan²α)/(1 + tan²α) = sin²α Step 1. Express tan²α in terms of sine and cosine:   tan²α = (sin²α)/(cos²α). Step 2. Substitute into the left‐hand side:   (sin²α/cos²α) / (1 + sin²α/cos²α). Step 3. Write 1 as cos²α/cos²α to combine the denominator:   = (sin²α/cos²α) / ((cos²α + sin²α)/cos²α). Step 4. Since cos²α + sin²α = 1, the expression simplifies to:   = (sin²α/cos²α) / (1/cos²α) = sin²α. Thus, the identity is established. ───────────────────────────── Conclusion Each identity follows directly by substituting definitions and using the Pythagorean identities:   • (1 + tan²x) = sec²x, with sec x = 1/cos x.   • sin²x + cos²x = 1.   • 1 − sin²x = cos²x. This completes the proofs of all the identities.