Is (1+csc(θ))(1-csc(θ))=-cot^2(θ) ?

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

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prove (1+csc(θ))(1-csc(θ))=-cot^2(θ)

prove

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

True

Solution steps

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

Manipulating left side

(1 + csc (θ)) (1 - csc (θ))

Expand

(1 + csc (θ)) (1 - csc (θ)) : 1 - csc^{2} (θ)

= 1 - csc^{2} (θ)

Use the Pythagorean identity:

csc^{2} (θ) = 1 + cot^{2} (θ)

csc^{2} (θ) - 1 = cot^{2} (θ)

= - cot^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

cot (u) tan (u) - sin^{2} (u) = cos^{2} (u)

\frac{1}{cos ( θ )} = \frac{tan ( θ )}{sin ( θ )}

(tan (y) + cot (y)) sin (y) cos (y) = 1

sin (x) \cdot cot (x) = cos (x)

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

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