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If cos^4 x / cos^2 y + sin^4 x / sin^2 y = 1 then prove that cos^4 y / cos^2 x + sin^4 y / sin^2 x=1.

If cos^4 x / cos^2 y + sin^4 x / sin^2 y = 1 then prove that cos^4 y / cos^2 x + sin^4 y / sin^2 x=1.

Grade:11

1 Answers

Aditya Gupta
2081 Points
4 years ago
the question is a tricky one for sure, but once you see how it is solved, there is nothing to it.
first start by multiplying cos^4 x / cos^2 y + sin^4 x / sin^2 y = 1 by sin^2y*cos^2y
then write sin^2y as 1-cos^2y.
it becomes cos^2(1-cos^2y)=cos^4x+cos^2y(sin^4x-cos^4x)=cos^4x+cos^2y(sin^2x-cos^2x)
or cos^4y+cos^4x-2cos^2xcos^2y=0
or (cos^2x-cos^2y)^2=0
or cos^2x=cos^2y
and sin^2x=sin^2y
Now, cos^4 y / cos^2 x + sin^4 y / sin^2 x=cos^2y+sin^2y=1 Hence proved

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