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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

cos (x)

= sin (x)

= sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (θ) + cos^{2} (θ) = 1

sec (θ) - cos (θ) = sin (θ) tan (θ)

sin (x - \frac{π}{2}) = - cos (x)

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

cos (x) tan (x) = sin (x)

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