Is 1-sin(θ)=(cos^2(θ))/(1+sin(θ)) ?

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

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prove 1-sin(θ)=(cos^2(θ))/(1+sin(θ))

prove

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

True

Solution steps

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

Manipulating left side

1 - sin (θ)

Multiply by

\frac{1 + sin ( θ )}{1 + sin ( θ )}

= \frac{( 1 - sin ( θ ) ) ( 1 + sin ( θ ) )}{1 + sin ( θ )}

Apply Difference of Two Squares Formula:

θ^{2} - y^{2} = (θ + y) (θ - y)

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

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

Use the Pythagorean identity:

1 = cos^{2} (θ) + sin^{2} (θ)

1 - sin^{2} (θ) = cos^{2} (θ)

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ( 2 x ) + cot ( 2 x )}{csc ( 2 x )} = sec (2 x)

\frac{cos ^{2} ( x )}{sin ( x )} = csc (x) - sin (x)

1 - 4 sin^{2} (x) cos^{2} (x) = cos^{2} (2 x)

sin (3 x) = sin (x) (4 cos^{2} (x) - 1)

sin^{2} (z) = \frac{1 - cos ( 2 z )}{2}

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