How do you prove that $cot^2 x +csc^2 x = 2csc^2 x - sin^2x-cos^2x$ ?

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Answer 1 · 2015-12-20T09:45:03

$csc^2(x) - sin^2(x) - cos^2(x) = 1/sin^2(x) - sin^2(x)-cos^2(x)$

$= 1/sin^2(x) - (sin^2(x) + cos^2(x))$

$= 1/sin^2(x) - 1$

$= 1/sin^2(x) - sin^2(x)/sin^2(x)$

$= (1-sin^2(x))/sin^2(x)$

$= cos^2(x)/sin^2(x)$

$= cot^2(x)$

Then, as

$cot^2(x)=csc^2(x) - sin^2(x) - cos^2(x)$

Adding $csc^2(x)$ to both sides gives us

$cot^2(x) + csc^2(x) = 2csc^2(x) - sin^2(x) - cos^2(x)$

How do you prove that cot^2 x +csc^2 x = 2csc^2 x - sin^2x-cos^2x cot^2 x +csc^2 x = 2csc^2 x - sin^2x-cos^2x ?