Is (csc(θ))/(cot(θ))=sec(θ) ?

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

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prove (csc(θ))/(cot(θ))=sec(θ)

prove

\frac{csc ( θ )}{cot ( θ )} = sec (θ)

True

Solution steps

\frac{csc ( θ )}{cot ( θ )} = sec (θ)

Manipulating left side

\frac{csc ( θ )}{cot ( θ )}

Express with sin, cos

\frac{csc ( θ )}{cot ( θ )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

sin (θ)

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( θ )}}

Simplify

\frac{1}{\frac{1}{s e c ( θ )}}

Apply the fraction rule:

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

= \frac{sec ( θ )}{1}

Apply rule

\frac{a}{1} = a

= sec (θ)

sec (θ)

sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( t )}{1 - cos ( t )} = csc (t) + cot (t)

sin (- x) + cot (- x) cos (- x) = - csc (x)

coth^{2} (x) - csch^{2} (x) = 1

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

csc (θ) (1 - cos^{2} (θ)) = sin (θ)

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