Prove that : 1/sin^2A -1/tan^2A =1

Question

asked Dec 26, 2021 21.7k views

1 Answer

Isxpjm · Answer 1 · 2022-01-01T02:46:01+0000

To prove that:

$$(1)/(\sin ^(2)A)-(1)/(\tan ^(2)A)=1$

LHS =
$(1)/(\sin ^(2)A)-(1)/(\tan ^(2)A)$

Using basic trigonometric identity,
$\tan (x)=(\sin (x))/(\cos (x))$

$$=(1)/(\sin ^(2)A)-(1)/(\left((\sin A)/(\cos A)\right)^(2))$

$$=(1)/(\sin ^(2)A)-(1)/((\sin^2A)/(\cos^2 A))$

$$=(1)/(\sin ^(2)A)-(\cos^2 A)/(\sin^2A)$

$$=\frac{1-{\cos^2 A}}{\sin ^(2)A}$

Using trigonometric identity:
$1-\cos ^(2)(x)=\sin ^(2)(x)$

$$=(\sin ^(2)A)/(\sin ^(2)A)$

= 1

= RHS

LHS = RHS

$$(1)/(\sin ^(2)A)-(1)/(\tan ^(2)A)=1$

Hence proved.