How do you prove $csc^4[theta]-cot^4[theta]=2csc^2-1$ ?

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Answer 1 · 2016-05-02T21:05:40

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Left Side: $=csc^4 theta - cot^4 theta$

$=1/sin^4 theta - cos^4 theta /sin^4 theta$

$=(1-cos^4 theta)/sin^4 theta$

$=((1+cos^2 theta)(1-cos^2 theta))/sin^4 theta$

$=((1+cos^2 theta)sin^2 theta)/sin^4 theta$

$=(1+cos^2 theta)/sin^2 theta$

$=1/sin^2 theta + cos^2 theta/sin^2 theta$

$=csc^2 theta +cot^2 theta$ ---> $cot^2 theta = csc^2 theta -1$

$=csc^2 theta+csc^2 theta -1$

$=2csc^2 theta -1$

$=$ Right Side

How do you prove csc^4[theta]-cot^4[theta]=2csc^2-1csc^4[theta]-cot^4[theta]=2csc^2-1?