Is 1/(tan(α))*1/(cos(α))= 1/(sin(α)) ?

The answer to whether 1/(tan(α))*1/(cos(α))= 1/(sin(α)) is True

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prove 1/(tan(α))*1/(cos(α))= 1/(sin(α))

prove

\frac{1}{tan ( α )} \cdot \frac{1}{cos ( α )} = \frac{1}{sin ( α )}

True

Solution steps

\frac{1}{tan ( α )} \cdot \frac{1}{cos ( α )} = \frac{1}{sin ( α )}

Manipulating left side

\frac{1}{tan ( α )} \frac{1}{cos ( α )}

Express with sin, cos

\frac{1}{cos ( α )} \cdot \frac{1}{tan ( α )}

Use the basic trigonometric identity:

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

= \frac{1}{cos ( α )} \cdot \frac{1}{\frac{s i n ( α )}{c o s ( α )}}

Simplify

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

\frac{1}{cos ( α )} \cdot \frac{1}{\frac{s i n ( α )}{c o s ( α )}}

Multiply fractions:

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

= \frac{1 \cdot 1}{cos ( α ) \frac{s i n ( α )}{c o s ( α )}}

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply

cos (α) \frac{sin ( α )}{cos ( α )} : sin (α)

cos (α) \frac{sin ( α )}{cos ( α )}

Multiply fractions:

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

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

Cancel the common factor:

cos (α)

= sin (α)

= \frac{1}{sin ( α )}

= \frac{1}{sin ( α )}

= \frac{1}{sin ( α )}

We showed that the two sides could take the same form

\Rightarrow True

Is 1/(tan(α))*1/(cos(α))= 1/(sin(α)) ?
The answer to whether 1/(tan(α))*1/(cos(α))= 1/(sin(α)) is True