Is (tan(θ))/(sec(θ))=sin(θ) ?

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

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prove (tan(θ))/(sec(θ))=sin(θ)

prove

\frac{tan ( θ )}{sec ( θ )} = sin (θ)

True

Solution steps

\frac{tan ( θ )}{sec ( θ )} = sin (θ)

Manipulating left side

\frac{tan ( θ )}{sec ( θ )}

Express with sin, cos

\frac{tan ( θ )}{sec ( θ )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

cos (θ)

= sin (θ)

= sin (θ)

= sin (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

(1 - sin^{2} (θ)) sec^{2} (θ) = 1

(1 + sin (a)) (1 - sin (a)) = cos^{2} (a)

sin^{3} (θ) = \frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)

tan^{2} (θ) + 1 = sec^{2} (θ)

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