Is (tan(θ))/(sin(θ))=sec(θ) ?

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

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prove (tan(θ))/(sin(θ))=sec(θ)

prove

\frac{tan ( θ )}{sin ( θ )} = sec (θ)

True

Solution steps

\frac{tan ( θ )}{sin ( θ )} = sec (θ)

Manipulating left side

\frac{tan ( θ )}{sin ( θ )}

Express with sin, cos

\frac{tan ( θ )}{sin ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

= \frac{sin ( θ )}{cos ( θ ) sin ( 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{tan ( 2 x )}{tan ( x )} = 1 + sec (2 x)

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

cot^{2} (x) + 1 = \frac{1}{sin ^{2} ( x )}

cos (2 θ) = cos^{4} (θ) - sin^{4} (θ)

\frac{sin ( 2 A )}{1 - cos ( 2 A )} = cot (A)

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