Is (csc(x)+cot(x))(csc(x)-cot(x))=1 ?

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

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prove (csc(x)+cot(x))(csc(x)-cot(x))=1

prove

(csc (x) + cot (x)) (csc (x) - cot (x)) = 1

True

Solution steps

(csc (x) + cot (x)) (csc (x) - cot (x)) = 1

Manipulating left side

(csc (x) + cot (x)) (csc (x) - cot (x))

Expand

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

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

Use the Pythagorean identity:

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

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

= 1

We showed that the two sides could take the same form

\Rightarrow True

cot^{2} (θ) - csc^{2} (θ) = - 1

\frac{sec ^{2} ( t ) - 1}{sec ^{2} ( t )} = sin^{2} (t)

cos (θ) = cos (- θ)

4 cos^{3} (x) - 3 cos (x) = cos (3 x)

\frac{csc ^{2} ( θ )}{cot ( θ )} = sec (θ) csc (θ)

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