Is (1-cos^2(θ))/(1-sin^2(θ))=tan^2(θ) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (1-cos^2(θ))/(1-sin^2(θ))=tan^2(θ)

prove

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

True

Solution steps

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

Manipulating right side

tan^{2} (θ)

Express with sin, cos

tan^{2} (θ)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= (\frac{sin ( θ )}{cos ( θ )})^{2}

Simplify

(\frac{sin ( θ )}{cos ( θ )})^{2} : \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

(\frac{sin ( θ )}{cos ( θ )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

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

Rewrite using trig identities

\frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

Use the Double Angle identity:

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

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

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

Use the Pythagorean identity:

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

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

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

Use the Double Angle identity:

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

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

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

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

Group like terms

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

Add similar elements:

cos^{2} (θ) - cos^{2} (θ) = 0

= - sin^{2} (θ) + 1

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

cos (3 x) = 4 (cos^{3} (x)) - 3 cos (x)

\frac{1 + csc ( x )}{sec ( x )} = cot (x) cos (x)

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

cot (θ) csc (θ) sec (θ) = csc^{2} (θ)

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