Is (csc(θ))/(sec(θ))=cot(θ) ?

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

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prove (csc(θ))/(sec(θ))=cot(θ)

prove

\frac{csc ( θ )}{sec ( θ )} = cot (θ)

True

Solution steps

\frac{csc ( θ )}{sec ( θ )} = cot (θ)

Manipulating left side

\frac{csc ( θ )}{sec ( θ )}

Express with sin, cos

\frac{csc ( θ )}{sec ( θ )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( θ )}}{sec ( θ )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( θ )}}{\frac{1}{c o s ( θ )}}

Simplify

\frac{\frac{1}{s i n ( θ )}}{\frac{1}{c o s ( θ )}} : \frac{cos ( θ )}{sin ( θ )}

\frac{\frac{1}{s i n ( θ )}}{\frac{1}{c o s ( θ )}}

Divide fractions:

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

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

Refine

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

Use the basic trigonometric identity:

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

= cot (θ)

We showed that the two sides could take the same form

\Rightarrow True

sec (x) csc (x) = tan (x) + cot (x)

sin (x) \cdot sec (x) = tan (x)

cot (B) sin (B) = cos (B)

\frac{cos ( x )}{sec ( x ) - tan ( x )} = 1 + sin (x)

\frac{csc ^{2} ( x )}{sec ^{2} ( x )} = csc^{2} (x) - 1

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