Is sec(θ)=(tan(θ))/(sin(θ)) ?

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

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prove sec(θ)=(tan(θ))/(sin(θ))

prove

sec (θ) = \frac{tan ( θ )}{sin ( θ )}

True

Solution steps

sec (θ) = \frac{tan ( θ )}{sin ( θ )}

Manipulating right 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{1 + tan ^{2} ( x )}{csc ^{2} ( x )} = tan^{2} (x)

sec (x) cos (x) = 1

sec (x) - cos (x) = \frac{tan ^{2} ( x )}{sec ( x )}

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

tan (\frac{x}{2}) = \frac{1 - cos ( x )}{sin ( x )}

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