Prove the identities. \( 1 \frac{\cos x}{1-\sin x}=\frac{1+\sin x}{\cos x} \) \( 2 \frac{\cos ^{2} x-\cos x-\sin ^{2} x}{2 \sin x \cdot \cos x+\sin x}=\frac{1}{\tan x}-\frac{1}{\sin x} \)

We wish to prove two trigonometric identities. We will prove each in turn. ────────────────────────────── Identity 1:   cos x/(1 – sin x) = (1 + sin x)/cos x Step 1. Start with the left-hand side and multiply numerator and denominator by (1 + sin x) (this is a common technique to “rationalize” the denominator):   cos x/(1 – sin x) = [cos x(1 + sin x)]/[(1 – sin x)(1 + sin x)] Step 2. Recognize that (1 – sin x)(1 + sin x) is a difference of two squares:   (1 – sin x)(1 + sin x) = 1 – sin²x Step 3. Using the Pythagorean identity 1 – sin²x = cos²x, we have:   cos x/(1 – sin x) = [cos x(1 + sin x)]/cos²x Step 4. Cancel a common factor of cos x (assuming cos x ≠ 0):   [cos x(1 + sin x)]/cos²x = (1 + sin x)/cos x Thus, we have shown:   cos x/(1 – sin x) = (1 + sin x)/cos x ────────────────────────────── Identity 2:   [cos²x – cos x – sin²x] / [2 sin x · cos x + sin x] = 1/ tan x – 1/ sin x Step 1. First, simplify the right-hand side. Recall that tan x = sin x/cos x, so:   1/tan x = cos x/sin x Thus:   1/ tan x – 1/ sin x = (cos x/sin x) – (1/sin x) = (cos x – 1)/ sin x Step 2. Now, we wish to show that:   [cos²x – cos x – sin²x] / [2 sin x cos x + sin x] = (cos x – 1)/ sin x Step 3. Factor the denominator. Notice that:   2 sin x · cos x + sin x = sin x (2 cos x + 1) Step 4. Write the identity we need to prove as:   [cos²x – cos x – sin²x] / [sin x (2 cos x + 1)] = (cos x – 1)/ sin x Multiply both sides by sin x (assuming sin x ≠ 0):   [cos²x – cos x – sin²x] / (2 cos x + 1) = cos x – 1 Step 5. Multiply both sides by (2 cos x + 1):   cos²x – cos x – sin²x = (cos x – 1)(2 cos x + 1) Step 6. Expand the right-hand side:   (cos x – 1)(2 cos x + 1) = [2 cos²x + cos x – 2 cos x – 1] = 2 cos²x – cos x – 1 Step 7. Hence, the equation becomes:   cos²x – cos x – sin²x = 2 cos²x – cos x – 1 Step 8. Subtract cos²x – cos x from both sides:   – sin²x = cos²x – 1 Step 9. Recognize that the right-hand side is equal to – sin²x by the Pythagorean identity cos²x – 1 = – sin²x. Thus, the equation holds true. ────────────────────────────── Conclusion: 1) cos x/(1 – sin x) = (1 + sin x)/cos x 2) [cos²x – cos x – sin²x] / [2 sin x cos x + sin x] = 1/ tan x – 1/ sin x Each identity has been verified using standard trigonometric identities and algebraic manipulations.