Solution assignment 01 Addition and subtraction formulas

Show:

$\sin (2x) = \displaystyle\frac{{2\tan (x)}}{{1 + \tan ^2 (x)}}$

In the right-hand side we substitute:

$\tan (x) = \displaystyle\frac{{\sin (x)}}{{\cos (x)}}$

and get:

$\displaystyle\frac{2\displaystyle\frac{\sin(x)}{\cos(x)}}{1+\displaystyle\frac{\sin^2(x)}{\cos^2(x)}}$

We multiply both numerator and denominator by $\cos^2(x)$ :

$\displaystyle\frac{{2\sin (x)\cos (x)}}{{\cos ^2 (x) + \sin ^2 (x)}} = \displaystyle\frac{{2\sin (x)\cos (x)}}{1} = \sin (2x)$