Is (cos(θ)-1)(cos(θ)+1)=-sin^2(θ) ?

The answer to whether (cos(θ)-1)(cos(θ)+1)=-sin^2(θ) is True

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prove (cos(θ)-1)(cos(θ)+1)=-sin^2(θ)

prove

(cos (θ) - 1) (cos (θ) + 1) = - sin^{2} (θ)

True

Solution steps

(cos (θ) - 1) (cos (θ) + 1) = - sin^{2} (θ)

Manipulating left side

(cos (θ) - 1) (cos (θ) + 1)

Expand

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

= cos^{2} (θ) - 1

Use the Pythagorean identity:

1 = cos^{2} (θ) + sin^{2} (θ)

1 - cos^{2} (θ) = sin^{2} (θ)

= - sin^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

\frac{csc ( β )}{cot ( β )} = sec (β)

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

\frac{csc ( x ) - cot ( x )}{1 - cos ( x )} = csc (x)

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

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