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

sin (4 5^{\circ}) cot (4 5^{\circ}) = cos (4 5^{\circ})

sec (4 x) = 2

\frac{1}{cos ^{2} ( A )} + \frac{1}{cot ^{2} ( A )} = 1

csc (θ) - sin (θ) = cot (θ) csc (θ)

cos (8 θ) = cos^{2} (4 θ) - sin^{2} (4 θ)

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