Question

sin theta -2 sin^3theta/2cos^3theta-costheta = tan theta

Answer 1

Hello,

Solution:

Note : using x instead of theta

$\frac{sin \: x - 2 {sin}^{3}x }{2 {cos}^{3} x - cos \: x} \\ \\ = \frac{sinx(1 - 2 {sin}^{2}x )}{cos \: x(2 {cos}^{2}x - 1) }$

As we know that

$cos \: 2x = 2 {cos}^{2}x - 1 \\ \\ and \: cos \: 2x = 1 - 2 {sin}^{2}x$

insert these two Formula's in the equation

$= \frac{sinx(1 - 2 {sin}^{2}x )}{cos \: x(2 {cos}^{2}x - 1) } \\ \\ = \frac{sin \: x \: cos \: 2x}{cos \: x \: cos \: 2x} \\ \\ = \frac{sin \: x}{cos \: x} \\ \\ = tan \: x$

is RHS.

Hence proved.

Hope it helps you.