Is sec(θ)cot(θ)=csc(θ) ?

The answer to whether sec(θ)cot(θ)=csc(θ) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove sec(θ)cot(θ)=csc(θ)

prove

sec (θ) cot (θ) = csc (θ)

True

Solution steps

sec (θ) cot (θ) = csc (θ)

Manipulating left side

sec (θ) cot (θ)

Express with sin, cos

cot (θ) sec (θ)

Use the basic trigonometric identity:

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

= \frac{cos ( θ )}{sin ( θ )} sec (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (θ)

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

\frac{1}{\frac{1}{c s c ( θ )}}

Simplify

\frac{1}{\frac{1}{c s c ( θ )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ( θ )}{1}

Apply rule

\frac{a}{1} = a

= csc (θ)

csc (θ)

csc (θ)

We showed that the two sides could take the same form

\Rightarrow True

cos (x - π) = - cos (x)

\frac{csc ^{2} ( θ )}{1 + tan ^{2} ( θ )} = cot^{2} (θ)

\frac{tan ( y )}{csc ( y )} = sec (y) - cos (y)

cos (3 x) = cos (2 x + x)

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