If 3 cos x ≠ 2 sin x, then find the general solution of sin2x – co

1.	If 3 cos x ≠ 2 sin x, then find the general solution of sin2x – cos 2x = 2 – sin 2x.
Answer» Given: sin²x – cos 2x = 2 – sin 2x ⇒ 1 – cos²x – (2cos²x – 1) = 2 – 2 sin x cos x ⇒ – 3 cos²x + 2 sin x cos x = 0 ⇒ cos x (2 sin x – 3 cos x) = 0 ⇒ cos x = 0 (∵ 2 sin x ≠ 3 cos x) ⇒ x = 2nπ ± \(\frac{\pi}{2}\) ⇒ x = (4n ± 1) \(\frac{\pi}{2}\)

If 3 cos x ≠ 2 sin x, then find the general solution of sin2x – cos 2x = 2 – sin 2x.

Answer»

Given:

sin²x – cos 2x = 2 – sin 2x

⇒ 1 – cos²x – (2cos²x – 1)

= 2 – 2 sin x cos x

⇒ – 3 cos²x + 2 sin x cos x = 0

⇒ cos x (2 sin x – 3 cos x) = 0 ⇒ cos x = 0 (∵ 2 sin x ≠ 3 cos x)

⇒ x = 2nπ ± \(\frac{\pi}{2}\)

⇒ x = (4n ± 1) \(\frac{\pi}{2}\)