Question

Prove that: cot(^2) x- csc x/csc(^2) x = cos(^2) x- sin x

Answer 1

$\frac{cot^2x -cosec\ x}{cosec^2x} = cos^2x - sin\ x$

Solution:

Given that we have to prove:

$\frac{cot^2x -cosec\ x}{cosec^2x} = cos^2x - sin\ x$

Take the L.H.S of above

$\frac{cot^2x -cosec\ x}{cosec^2x}$

Simplify

$\frac{cot^2x}{cosec^2x} -\frac{cosec\ x}{cosec^2x}\\\\Simplify\\\\\frac{cot^2x}{cosec^2x} -\frac{1}{cosec\ x}$

We know that,

$sin\ x = \frac{1}{cosec\ x}$

Therefore,

$\frac{cot^2x}{cosec^2x} -sin\ x$

Also we know that,

$cot^2x = \frac{cos^2\ x}{sin^2\ x}$

$cosec^2\ x = \frac{1}{sin^2\ x}$

Thus we get,

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

Thus,

L.H.S = R.H.S

Thus proved