Question

Prove that sinx -2sin^3x/2cos^3x- cosx =tanx

Answer 1

Step-by-step explanation:

To prove

$\frac{sin \: x - 2 {sin}^{3}x }{2 {cos}^{3}x - cos \: x } = tan \: x$

Take LHS

$\frac{sin \: x (1- 2 {sin}^{2}x )}{cos \: x(2 {cos}^{2}x - 1)} \\ \\ as \: we \: know \: that \\ \\ {sin}^{2}x = 1 - {cos}^{2}x \\ \\ put \: this \: in \: the \: numerator \\ \\ \frac{sin \: x (1- 2 (1 - {cos}^{2}x ))}{cos \: x(2 {cos}^{2}x - 1)} \\ \\ = \frac{sin \: x (1- 2 + 2{cos}^{2}x )}{cos \: x(2 {cos}^{2}x - 1)} \\ \\ = \frac{sin \: x ( 2{cos}^{2}x - 1)}{cos \: x(2 {cos}^{2}x - 1)} \\ \\ = \frac{sin \: x}{cos \: x} \\ \\ = tan \: x$

= RHS

hence proved

Hope it helps you.

Answer 2

$\frac{\sin \: x - 2 {\sin}^{3}x }{2 {\cos}^{3}x - \cos \: x }=\tan x$ Proved.

Step-by-step explanation:

Consider the provided information.

$\frac{\sin \: x - 2 {\sin}^{3}x }{2 {\cos}^{3}x - \cos \: x }=\tan x$

Consider the LHS of the above equation.

Take sin(x) and cos(x) common.

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

Use the identity: sin²θ=1-cos²θ

$\frac{\sin x[1-2(1-\cos^2x)]}{\cos x(2\cos^2x-1)}$

Use the identity: $\frac{\sin x}{\cos x}=\tan x$

$\tan x\times\frac{1-2+2\cos^2x}{2\cos^2x-1}$

$\tan x\times\frac{2\cos^2x-1}{2\cos^2x-1}$

$\tan x$

Hence, proved

#Learn more

Cos theta/1-tan theta + sin theta/1-cot theta

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