Question

verify the identity.

\tan 2 alpha = \frac { 2 \tan alpha } { 1 - \tan ^ { 2 } alpha }

to verify the identity, start with the left side and transform it to look like the right side. choose the correct transformations and transform the expression at each step.

\tan 2 alpha = \tan

Explanation:

Step1: Use the tangent addition formula

The formula for \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\). Let \(A = B=\alpha\), then \(\tan(2\alpha)=\tan(\alpha+\alpha)\).

Step2: Substitute into the formula

Substitute \(A = B=\alpha\) into \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\), we get \(\tan(2\alpha)=\frac{\tan\alpha+\tan\alpha}{1 - \tan\alpha\tan\alpha}\).

Step3: Simplify the expression

Simplify \(\frac{\tan\alpha+\tan\alpha}{1 - \tan\alpha\tan\alpha}\) to \(\frac{2\tan\alpha}{1-\tan^{2}\alpha}\).

Answer:

\(\tan2\alpha=\frac{2\tan\alpha}{1 - \tan^{2}\alpha}\) is a verified identity.