Question

Prove that , (cos2

A-1) (1+ cot

2

A) = -1​

Answer 1

Question:

Prove that:

$\displaystyle{\sf\:(\:\cos^2\:A\:-\:1\:)\:(\:1\:+\:\cot^2\:A\:)\:=\:-\:1}$

Answer:

$\displaystyle{\boxed{\red{\sf\:(\:\cos^2\:A\:-\:1\:)\:(\:1\:+\:\cot^2\:A\:)\:=\:-\:1\:}}}$

Step-by-step-explanation:

We have given a trigonometric equation.

We have to prove that equation.

The given trigonometric equation is

$\displaystyle{\sf\:(\:\cos^2\:A\:-\:1\:)\:(\:1\:+\:\cot^2\:A\:)\:=\:-\:1}$

We know that,

$\displaystyle{\boxed{\blue{\sf\:1\:+\:\cot^2\:A\:=\:cosec^2\:A\:}}}$

$\displaystyle{\implies\sf\:LHS\:=\:(\:\cos^2\:A\:-\:1\:)\:cosec^2\:A}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\sin^2\:A\:+\:\cos^2\:A\:=\:1\:}}}$

$\displaystyle{\therefore\:\sf\:\cos^2\:A\:=\:1\:-\:\sin^2\:A}$

$\displaystyle{\implies\sf\:LHS\:=\:(\:\cancel{1}\:-\:\sin^2\:A\:-\:\cancel{1}\:)\:cosec^2\:A}$

$\displaystyle{\implies\sf\:LHS\:=\:(\:-\:\sin^2\:A\:)\:cosec^2\:A}$

We know that,

$\displaystyle{\boxed{\green{\sf\:cosec\:A\:=\:\dfrac{1}{\sin\:A}\:}}}$

$\displaystyle{\implies\sf\:LHS\:=\:(\:-\:\sin^2\:A\:)\:\left(\:\dfrac{1}{\sin\:A}\:\right)^2}$

$\displaystyle{\implies\sf\:LHS\:=\:-\:\cancel{\sin^2\:A}\:\times\:\dfrac{1}{\cancel{\sin^2\:A}}}$

$\displaystyle{\implies\sf\:LHS\:=\:-\:1}$

$\displaystyle{\sf\:RHS\:=\:-\:1}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:LHS\:=\:RHS\:}}}}$

Hence proved!