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verifing an identity algebraically

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csc^4x-2csc^2x+1=cot^4x

Apr 9th, 2015

csc^4 x − 2csc^2 x + 1 = cot^4 x

(1/sin x)^4 − 2(1/sin x)^2 + 1 = (cos x/sin x)^4

Multiplying by sin^4 x on both sides, we get

(1/sin x)^4 − 2(1/sin x)^2 + 1 = (cos x/sin x)^4

1 - 2sin ^2 x + sin^4 x = cos^4 x

1 - 2(1 - cos^2 x) + (1 - cos^2 x)^2 = cos^4 x

1 - 2 - 2cos^2 x + 1 + cos^4 x - 2cos^2 x = cos^4 x

2 - 2 + 2cos^2 x - 2cos^2 x + cos^4 x = cos^4 x

cos^4 x = cos^4 x

By dividing by sin^4 x on both sides, we get

cos^4 x / sin^4 x = cos^4 x / sin^4 x

cot^4 x = cot^4 x

L.H.S = R.H.S

Hence proved

Apr 9th, 2015

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