Question

Identity transformations of Trigonometric Expressions.
prove the following identities.
$\frac{sinx+cosx}{co{s}^3}=ta{n}^{3}x+ta{n}^{2}x+tanx+1.$

Answer 1

$\frac{\sin x+\cos x}{{\cos }^{3}x}$

$=\frac{\sin x}{{\cos }^{3}x}+\frac{\cos x}{{\cos }^{3}x}$

$=\frac{\sin x}{\cos x}\times \frac{1}{{\cos }^{2}x}+\frac{1}{{\cos }^{2}x}$

$=\tan x.{\sec }^{2}x+{\sec }^{2}x$

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

$=\tan x+{\tan }^{3}x+1+{\tan }^{2}x$

$={\tan }^{3}x+{\tan }^{2}x+\tan x+1$

=RHS=Proved