Is (sec(θ))/(tan(θ))=csc(θ) ?

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

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prove (sec(θ))/(tan(θ))=csc(θ)

prove

\frac{sec ( θ )}{tan ( θ )} = csc (θ)

True

Solution steps

\frac{sec ( θ )}{tan ( θ )} = csc (θ)

Manipulating left side

\frac{sec ( θ )}{tan ( θ )}

Express with sin, cos

\frac{sec ( θ )}{tan ( θ )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Refine

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

Cancel the common factor:

cos (θ)

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( θ )}}

Simplify

\frac{1}{\frac{1}{c s c ( θ )}}

Apply the fraction rule:

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

= \frac{csc ( θ )}{1}

Apply rule

\frac{a}{1} = a

= csc (θ)

csc (θ)

csc (θ)

We showed that the two sides could take the same form

\Rightarrow True

cot (2 b) = \frac{cot ^{2} ( b ) - 1}{2 cot ( b )}

1 + tan^{2} (a) = sec^{2} (a)

2 cos (x) cos (y) = cos (x + y) + cos (x - y)

\frac{1}{sec ^{2} ( θ )} + \frac{1}{csc ^{2} ( θ )} = 1

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

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