# Precalculus

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Verify.

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

• Precalculus -

Not a surprise that you don't get it,
the two expressions are not equivalent!

Start from left:
csc^4(x)-cot^4(x)
Factor sin^4(x) as denominator:
=(1-cos^4(x))/sin^4(x)
Factor as difference of two squares
= (1 + cos²(x)) (1 - cos²(x)) / sin^4(x)
=(1 + cos²(x))sin²(x)/sin^4(x)
=(1+cos²(x))/sin²(x)
=(2cos²(x)+sin²(x))/sin²(x)
=2cos²(x)/sin²(x)+1
=2cot²(x)+1
(and not 2csc²(x)+1)

• Precalculus -

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

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