Prove follaring identity. \( \frac{\cos (2 x+x)}{\cos x}=\{\cos 2 x-1 \)

We wish to show that   cos(2x + x)   —————— = 2 cos 2x – 1             (∗)    cos x for every x such that cos x ≠ 0. (Any other form, such as “cos2x – 1” on the right‐side, would be equivalent to 2 cos 2x – 1 only for special angles.) One way to prove (∗) is to use two well‐known trigonometric identities: 1. The cosine addition formula:   cos(A + B) = cos A cos B – sin A sin B. 2. The double‐angle formula for cosine:   cos 2x = 2 cos²x – 1. Also, it is known that the triple‐angle formula for cosine can be written as   cos 3x = 4 cos³x – 3 cos x. Below are two proofs of (∗). ───────────────────────────── Method 1. Starting from the triple‐angle formula We know that   cos 3x = 4 cos³x – 3 cos x. Dividing both sides by cos x (assuming cos x ≠ 0) gives   cos 3x / cos x = 4 cos²x – 3. But, using the double‐angle formula, note that   2 cos 2x – 1 = 2(2 cos²x – 1) – 1 = 4 cos²x – 2 – 1 = 4 cos²x – 3. Thus,   cos 3x / cos x = 2 cos 2x – 1. Since cos(2x + x) = cos 3x, we have proved (∗). ───────────────────────────── Method 2. Using the angle‐addition formula Step 1. Write cos(2x + x) by the cosine addition formula:   cos(2x + x) = cos 2x cos x – sin 2x sin x. Step 2. Divide both sides by cos x:   cos(2x + x) / cos x = cos 2x – (sin 2x sin x)/cos x. Step 3. Express sin 2x in terms of sin x and cos x by the double‐angle formula for sine:   sin 2x = 2 sin x cos x, so that   (sin 2x sin x) / cos x = (2 sin x cos x sin x) / cos x = 2 sin²x. Thus the expression becomes:   cos(2x + x) / cos x = cos 2x – 2 sin²x. Step 4. Replace sin²x by 1 – cos²x (using sin²x + cos²x = 1):   cos(2x + x) / cos x = cos 2x – 2(1 – cos²x)                 = cos 2x – 2 + 2 cos²x. Step 5. Now rewrite cos 2x in terms of cos²x by the identity   cos 2x = 2 cos²x – 1. Substitute to obtain:   cos(2x + x) / cos x = (2 cos²x – 1) – 2 + 2 cos²x                 = 4 cos²x – 3. Step 6. Notice that, as in Method 1,   2 cos 2x – 1 = 2(2 cos²x – 1) – 1 = 4 cos²x – 3. Thus,   cos(2x + x) / cos x = 2 cos 2x – 1, which is the desired identity. ───────────────────────────── Conclusion Both methods show that for all x with cos x ≠ 0,   cos(2x + x) / cos x = 2 cos 2x – 1. This completes the proof.