Is (cos^2(θ))/(1+sin(θ))=1-sin(θ) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (cos^2(θ))/(1+sin(θ))=1-sin(θ)

prove

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

True

Solution steps

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

Manipulating right 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

tan^{2} (x) + \frac{1}{sin ( x ) csc ( x )} = sec^{2} (x)

csc (- x) - sin (- x) = cos (- x) cot (- x)

(csc (x)) (tan (x)) = sec (x)

cot (\frac{x}{2}) = \frac{1 + cos ( x )}{sin ( x )}

(sin (x)) (csc (x)) = 1

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