Prove the identity. (sin 2x)/(1+cos 2x)=tan x

Question

Bytelearn · Accepted Answer

Apply Double Angle Formulas: We will start by using the double angle formulas for sine and cosine. The double angle formula for sine is sin 2x = 2 sin x cos x, and for cosine is cos 2x = cos^2 x - sin^2 x. We will apply these formulas to the left side of the identity.Rewrite sin 2x: Rewrite sin 2x using the double angle formula: sin 2x = 2 sin x cos x.Rewrite 1 + cos 2x: Rewrite 1 + cos 2x using the double angle formula and the Pythagorean identity sin^2 x + cos^2 x = 1: 1 + cos 2x = 1 + (cos^2 x - sin^2 x) = (1 - sin^2 x) + cos^2 x = cos^2 x + sin^2 x.Substitute into Identity: Now, we substitute the expressions we found into the left side of the identity: (sin 2x)/(1 + cos 2x) = (2 sin x cos x) / (cos^2 x + sin^2 x).Simplify Denominator: Since cos^2 x + sin^2 x = 1, we can simplify the denominator: (2 sin x cos x) / (cos^2 x + sin^2 x) = (2 sin x cos x) / 1 = 2 sin x cos x.Divide by cos x: Now, we divide both the numerator and the denominator by cos x, assuming cos x is not equal to zero (since division by zero is undefined): (2 sin x cos x) / cos x = 2 sin x.Correct Mistake: We realize there is a mistake in the previous step. We should have divided by cos^2 x, not cos x. Let's correct this: (2 sin x cos x) / cos^2 x = (2 sin x) / cos x.

Prove the identity. $\newline$ $\frac{\sin 2x}{1+\cos 2x}=\tan x$

Full solution

More problems from Find derivatives of other trigonometric functions

Related topics

Prove the identity.\newline sin⁡2x1+cos⁡2x=tan⁡x\frac{\sin 2x}{1+\cos 2x}=\tan x1+cos2xsin2x​=tanx

Prove the identity. $\newline$ $\frac{\sin 2x}{1+\cos 2x}=\tan x$