Question

verify the identity

(2tanx/1+tan^2(x)) = sin2x

Answer 1

\(\bf {\color{blue}{ 1+tan^2(\theta)=sec^2(\theta)\qquad \qquad sec(\theta)=\cfrac{1}{cos(\theta)}}}\\ \quad \\ \quad \\

\cfrac{2tan(x)}{1+tan^2(x)}\implies \cfrac{2tan(x)}{sec^2(x)}\implies
\cfrac{\frac{2sin(x)}{cos(x)}}{\frac{1}{cos^2(x)}}\)

recall your pythagorean identities

Answer 2

\[\frac{ 2tanx }{ 1+\tan^2x } = \frac{ 2\frac{ sinx }{ cosx } }{ \frac{ 1 }{ \cos^2x } } = 2\frac{ sinx }{ cosx } \times \cos^2x=2sinx.cosx = \sin 2x \]