Question

if 1+tan^2 5A= cosec^2 4A then A =

​

Answer 1

Answer:

The measure of angle A is 10°.

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:1\:+\:\tan^2\:(\:5\:A\:)\:=\:cosec^2\:(\:4\:A\:)}$

We have to find measure of angle A.

Now,

$\displaystyle{\sf\:1\:+\:\tan^2\:(\:5\:A\:)\:=\:cosec^2\:(\:4\:A\:)}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\sec^2\:(\:A\:)\:=\:1\:+\:\tan^2\:(\:A\:)\:}}}$

$\displaystyle{\implies\sf\:\sec^2\:(\:5\:A\:)\:=\:cosec^2\:(\:4\:A\:)}$

By taking square roots on both sides, we get,

$\displaystyle{\implies\sf\:\sec\:(\:5\:A\:)\:=\:cosec\:(\:4\:A\:)}$

We know that,

$\displaystyle{\boxed{\blue{\sf\:cosec\:(\:A\:)\:=\:\sec\:(\:90^{\circ}\:-\:A\:)\:}}}$

$\displaystyle{\implies\sf\:\sec\:(\:5\:A\:)\:=\:\sec\:(\:90^{\circ}\:-\:4\:A\:)}$

By taking sec inverse on both sides, we get,

$\displaystyle{\implies\sf\:\sec^{-\:1}\:(\:\sec\:(\:5\:A\:)\:)\:=\:\sec^{-\:1}\:(\:\sec\:(\:90^{\circ}\:-\:4\:A\:)\:)}$

$\displaystyle{\implies\sf\:5\:A\:=\:90^{\circ}\:-\:4\:A}$

$\displaystyle{\implies\sf\:5\:A\:+\:4\:A\:=\:90^{\circ}}$

$\displaystyle{\implies\sf\:9\:A\:=\:90^{\circ}}$

$\displaystyle{\implies\sf\:A\:=\:\dfrac{\cancel{90}}{\cancel{9}}}$

$\displaystyle{\implies\:\underline{\boxed{\red{\sf\:A\:=\:10^{\circ}\:}}}}$

∴ The measure of angle A is 10°.