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#