Is cos(θ)sec(θ)=tan(θ)cot(θ) ?

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

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prove cos(θ)sec(θ)=tan(θ)cot(θ)

prove

cos (θ) sec (θ) = tan (θ) cot (θ)

True

Solution steps

cos (θ) sec (θ) = tan (θ) cot (θ)

Manipulating left side

cos (θ) sec (θ)

Express with sin, cos

cos (θ) sec (θ)

Use the basic trigonometric identity:

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

= cos (θ) \frac{1}{cos ( θ )}

cos (θ) \frac{1}{cos ( θ )} = 1

cos (θ) \frac{1}{cos ( θ )}

Multiply fractions:

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

= \frac{1 \cdot cos ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= 1

= 1

= 1

Manipulating right side

tan (θ) cot (θ)

Express with sin, cos

cot (θ) tan (θ)

Use the basic trigonometric identity:

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

= \frac{cos ( θ )}{sin ( θ )} tan (θ)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (θ)

= \frac{sin ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

Is cos(θ)sec(θ)=tan(θ)cot(θ) ?
The answer to whether cos(θ)sec(θ)=tan(θ)cot(θ) is True