Is (cot(θ))/(csc(θ))=cos(θ) ?

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

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prove (cot(θ))/(csc(θ))=cos(θ)

prove

\frac{cot ( θ )}{csc ( θ )} = cos (θ)

True

Solution steps

\frac{cot ( θ )}{csc ( θ )} = cos (θ)

Manipulating left side

\frac{cot ( θ )}{csc ( θ )}

Express with sin, cos

\frac{cot ( θ )}{csc ( θ )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

sin (θ)

= cos (θ)

= cos (θ)

= cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

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

tan (x) + cot (x) = 2 csc (2 x)

cos (x + \frac{π}{2}) = - sin (x)

csc^{2} (x) tan (x) - tan (x) = cot (x)

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