Question

(tan a + cot a )² - (tan a - cot )² =4​

Answer 1

bro this is soo easy.. put it on a+b whole square and a-b whole square then tan a and tand b will got lapsed and u will got amswer 4

Answer 2

Answer:

$\displaystyle{\boxed{\red{\sf\:(\:\tan\:a\:+\:\cot\:a\:)^2\:-\:(\:\tan\:a\:-\:\cot\:a\:)^2\:=\:4}}}$

Step-by-step-explanation:

We have given a trigonometric equation.

We have to prove that equation.

The given trigonometric equation is

$\displaystyle{\sf\:(\:\tan\:a\:+\:\cot\:a\:)^2\:-\:(\:\tan\:a\:-\:\cot\:a\:)^2\:=\:4}$

Let,

$\displaystyle{\bullet\sf\:(\:\tan\:a\:+\:\cot\:a\:)\:=\:x}$

$\displaystyle{\bullet\sf\:(\:\tan\:a\:-\:\cot\:a\:)\:=\:y}$

$\displaystyle{\implies\sf\:x^2\:-\:y^2\:=\:4}$

$\displaystyle{\implies\sf\:LHS\:=\:x^2\:-\:y^2}$

$\displaystyle{\implies\sf\:LHS\:=\:(\:x\:+\:y\:)\:(\:x\:-\:y\:)}$

$\displaystyle{\implies}\displaystyle\sf\:LHS\:=\:[\:(\:\tan\:a\:+\:\cot\:a\:)\:+\:(\:\tan\:a\:-\:\cot\:a\:)\:]\:[\:(\:\tan\:a\:+\:\cot\:a\:)\:-\:(\:\tan\:a\:-\:\cot\:a\:)\:]\:\:\dots\:[\:Re-substituting\:]$

$\displaystyle{\implies\sf\:LHS\:=\:(\:\tan\:a\:+\:\cancel{\cot\:a}\:+\:\tan\:a\:-\:\cancel{\cot\:a}\:)\:(\:\cancel{\tan\:a}\:+\:\cot\:a\:-\:\cancel{\tan\:a}\:+\:\cot\:a\:)}$

$\displaystyle{\implies\sf\:LHS\:=\:(\:\tan\:a\:+\:\tan\:a\:)\:(\:\cot\:a\:+\:\cot\:a\:)}$

$\displaystyle{\implies\sf\:LHS\:=\:2\:\tan\:a\:\times\:2\:\cot\:a}$

$\displaystyle{\implies\sf\:LHS\:=\:4\:\cancel{\tan\:a}\:\times\:\dfrac{1}{\cancel{\tan\:a}}}$

$\displaystyle{\implies\sf\:LHS\:=\:4}$

$\displaystyle{\implies\sf\:RHS\:=\:4}$

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

Hence proved!