Question

Prove that :
1/sin^2A -1/tan^2A =1​

Answer 1

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.