Is (tan(β))/(sec(β))=sin(β) ?

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

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prove (tan(β))/(sec(β))=sin(β)

prove

\frac{tan ( β )}{sec ( β )} = sin (β)

True

Solution steps

\frac{tan ( β )}{sec ( β )} = sin (β)

Manipulating left side

\frac{tan ( β )}{sec ( β )}

Express with sin, cos

\frac{tan ( β )}{sec ( β )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

cos (β)

= sin (β)

= sin (β)

= sin (β)

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

(1 - cos (x)) (1 + sec (x)) = - 1

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

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