Is (1-cos(x))(1+cos(x))=sin^2(x) ?

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

prove (1-cos(x))(1+cos(x))=sin^2(x)

prove

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

True

Solution steps

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

Manipulating left side

(1 - cos (x)) (1 + cos (x))

Expand

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

(1 + cos (x)) (1 - cos (x))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = cos (x)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - cos^{2} (x)

= 1 - cos^{2} (x)

Use the Pythagorean identity:

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

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

= sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e cot (2 θ) = \frac{cot ^{2} ( θ ) - 1}{2 cot ( θ )}

p ro v e (sec^{2} (x) - 1) cos (x) = sin^{2} (x) sec (x)

p ro v e \frac{2}{cot ( x ) + tan ( x )} = sin (2 x)

p ro v e (1 - cos (θ)) (1 + cos (θ)) = sin^{2} (θ)

p ro v e \frac{1 - 3 cos ( x ) - 4 cos ^{2} ( x )}{sin ^{2} ( x )} = \frac{1 - 4 cos ( x )}{1 - cos ( x )}

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