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

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Answer 1 · 2018-03-18T22:12:37

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

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:
$sec^2x=sec^2x$