Question

Prove that sina-2sin^3a/2cos^3a-cosa=tana

Answer 1

sin a - 2 sin³a / 2 cos³a - cos a  =  tan a
________________________________________________________

Taking LHS

= sin a - 2 sin³a / 2 cos³a - cos a

= sin a (1-2 sin²a) / cos a (2 cos²a - 1)

= sin a (1-sin²a-sin²a) / cos a (cos²a+cos²a-1)

Then (1-sin²a = cos²a) and (cos²a-1 = -sin²a)

= sin a (cos²a-sin²a) / cos a (cos²a-sin²a)

Canceling (cos²a-sin²a) on both numerator and denominator.

= sin a / cos a

= tan a = RHS
_______________________________________________________

☺ ☺ ☺ Hope this Helps ☺ ☺ ☺

Answer 2

Answer:

$\frac{sina - 2 {sin}^{3} a}{2 {cos}^{3} a - cosa} \\ \\ = \frac{sina(1 - 2 {sin}^{2} a)}{cosa(2 {cos}^{2}a - 1) } \\ \\ = \frac{sina(cos2a)}{cosa(cos2a)} \\ \\ = \frac{sina}{cosa} \\ \\ = tana$