Is sin(θ)cot(θ)sec(θ)=1 ?

The answer to whether sin(θ)cot(θ)sec(θ)=1 is True

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prove sin(θ)cot(θ)sec(θ)=1

prove

sin (θ) cot (θ) sec (θ) = 1

True

Solution steps

sin (θ) cot (θ) sec (θ) = 1

Manipulating left side

sin (θ) cot (θ) sec (θ)

Express with sin, cos

cot (θ) sec (θ) sin (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (θ)

= \frac{1 \cdot 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