Question

How can I prove the following equation is an identity? 1+sec^2(x)sin^2(x)=sec^2(x)

Answer 1

`1+sec^2(x)sin^2(x)=sec^2(x)`

Use reciprocal identity:
`sec^2x=1/cos^2x`

Therefore:
`1+1/cos^2x*sin^2(x)=sec^2(x)`
`1+ sin^2x /cos^2x=sec^2(x)`

Use the quotient identity:
`sin^2x/cos^2x= tan^2x`

Therefore:
`1+ tan^2x=sec^2(x)`

Use the Pythagorean identity:
`1+ tan^2x=sec^2(x)`

Therefore:
`sec^2x=sec^2x`