Is csc^4(x)-cot^4(x)=1+2cot^2(x) ?

The answer to whether csc^4(x)-cot^4(x)=1+2cot^2(x) is True

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prove csc^4(x)-cot^4(x)=1+2cot^2(x)

prove

csc^{4} (x) - cot^{4} (x) = 1 + 2 cot^{2} (x)

True

Solution steps

csc^{4} (x) - cot^{4} (x) = 1 + 2 cot^{2} (x)

Manipulating left side

csc^{4} (x) - cot^{4} (x)

Factor

- cot^{4} (x) + csc^{4} (x) : (csc^{2} (x) + cot^{2} (x)) (csc (x) + cot (x)) (csc (x) - cot (x))

= (- cot (x) + csc (x)) (cot (x) + csc (x)) (cot^{2} (x) + csc^{2} (x))

Rewrite using trig identities

= 2 cot^{2} (x) + 1

= 1 + 2 cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1}{sin ( x )} sec (x) = tan (x) + cot (x)

\frac{1 - tan ^{2} ( x )}{cot ^{2} ( x ) - 1} = cos^{2} (x)

csc (A) - sin (A) = cot (A) cos (A)

cos (x) sec (x) + tan (x) = 1 + sin (x)

\frac{cos ( θ )}{sec ( θ ) - tan ( θ )} = 1 + sin (x)

Is csc^4(x)-cot^4(x)=1+2cot^2(x) ?
The answer to whether csc^4(x)-cot^4(x)=1+2cot^2(x) is True