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

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

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

prove

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

True

Solution steps

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

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

We showed that the two sides could take the same form

\Rightarrow True

(1 - sin^{2} (x)) tan^{2} (x) = sin^{2} (x)

(cos (x) - sin (x))^{2} = 1 - 2 sin (x) cos (x)

cos (- x) - sin (- x) = cos (x) + sin (x)

cos (θ) sec (θ) = 1

cos (a + b) + cos (a - b) = 2 cos (a) cos (b)

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