Is sec(x)=(tan(x))/(sin(x)) ?

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

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prove sec(x)=(tan(x))/(sin(x))

prove

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

True

Solution steps

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

Manipulating right side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

sin (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( x )}}

Simplify

\frac{1}{\frac{1}{s e c ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

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

tan (x) cot (x) sin (x) = csc (x)

\frac{7}{tan ( x )} + \frac{7}{cot ( x )} = 7 tan (x) + 7 cot (x)

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

4 tan (x) + 2 sin (x) cos (x) = 0

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