Is (cos^2(α))/(1-sin(α))=1+sin(α) ?

The answer to whether (cos^2(α))/(1-sin(α))=1+sin(α) 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(α))/(1-sin(α))=1+sin(α)

prove

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

True

Solution steps

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

Manipulating left side

\frac{cos ^{2} ( α )}{1 - sin ( α )}

Rewrite using trig identities

\frac{cos ^{2} ( α )}{1 - sin ( α )}

Use the Pythagorean identity:

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

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

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

Simplify

\frac{1 - sin ^{2} ( α )}{1 - sin ( α )} : sin (α) + 1

\frac{1 - sin ^{2} ( α )}{1 - sin ( α )}

Factor

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

1 - sin^{2} (α)

Factor out common term

- 1

= - (sin^{2} (α) - 1)

Factor

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

sin^{2} (α) - 1

Rewrite

1

1^{2}

= sin^{2} (α) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (α) + 1) (sin (α) - 1)

= - (sin (α) + 1) (sin (α) - 1)

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

Cancel

- \frac{( sin ( α ) + 1 ) ( sin ( α ) - 1 )}{1 - sin ( α )} : sin (α) + 1

- \frac{( sin ( α ) + 1 ) ( sin ( α ) - 1 )}{1 - sin ( α )}

- sin (α) + 1 = - (sin (α) - 1)

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

Refine

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

Cancel the common factor:

sin (α) - 1

= sin (α) + 1

= sin (α) + 1

= sin (α) + 1

= sin (α) + 1

= 1 + sin (α)

We showed that the two sides could take the same form

\Rightarrow True

2 sin^{2} (θ) - 1 = 1 - 2 cos^{2} (θ)

cos (θ - π) = - cos (θ)

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

csc (A - B) = \frac{csc ( A ) csc ( B )}{cot ( B ) - cot ( A )}

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

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