Is (sin(θ))/(cos^2(θ))=tan(θ)sec(θ) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (sin(θ))/(cos^2(θ))=tan(θ)sec(θ)

prove

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

True

Solution steps

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

Manipulating right side

tan (θ) sec (θ)

Express with sin, cos

sec (θ) tan (θ)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( θ )} tan (θ)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

Simplify

\frac{1}{cos ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )} : \frac{sin ( θ )}{cos ^{2} ( θ )}

\frac{1}{cos ( θ )} \cdot \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot sin ( θ )}{cos ( θ ) cos ( θ )}

Multiply:

1 \cdot sin (θ) = sin (θ)

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

cos (θ) cos (θ) = cos^{2} (θ)

cos (θ) cos (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= cos^{2} (θ)

= \frac{sin ( θ )}{cos ^{2} ( θ )}

= \frac{sin ( θ )}{cos ^{2} ( θ )}

= \frac{sin ( θ )}{cos ^{2} ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

tan^{2} (u) (1 + cos (2 u)) = 1 - cos (2 u)

\frac{sec ( v ) - cos ( v )}{sec ( v )} = sin^{2} (v)

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

sec^{2} (θ) - tan (θ) cot (θ) = tan^{2} (θ)

csc^{2} (θ) (1 - cos^{2} (θ)) = tan (42 0^{\circ})

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