Is (1+sin(θ))(1-sin(θ))=cos^2(θ) ?

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

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prove (1+sin(θ))(1-sin(θ))=cos^2(θ)

prove

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

True

Solution steps

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

Manipulating left side

(1 + sin (θ)) (1 - sin (θ))

Expand

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

(1 + sin (θ)) (1 - sin (θ))

Apply Difference of Two Squares Formula:

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

a = 1, b = sin (θ)

= 1^{2} - sin^{2} (θ)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (θ)

= 1 - sin^{2} (θ)

Use the Pythagorean identity:

1 = cos^{2} (θ) + sin^{2} (θ)

1 - sin^{2} (θ) = cos^{2} (θ)

= cos^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

sin (θ) (tan (θ) + cot (θ)) = sec (θ)

cot (2 θ) = \frac{cot ^{2} ( θ ) - 1}{2 cot ( θ )}

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

sin^{2} (θ) = 1 - cos^{2} (θ)

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

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