Is cot(α)sec(α)=csc(α) ?

The answer to whether cot(α)sec(α)=csc(α) is True

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prove cot(α)sec(α)=csc(α)

prove

cot (α) sec (α) = csc (α)

True

Solution steps

cot (α) sec (α) = csc (α)

Manipulating left side

cot (α) sec (α)

Express with sin, cos

cot (α) sec (α)

Use the basic trigonometric identity:

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

= \frac{cos ( α )}{sin ( α )} sec (α)

Use the basic trigonometric identity:

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

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

Simplify

\frac{cos ( α )}{sin ( α )} \cdot \frac{1}{cos ( α )} : \frac{1}{sin ( α )}

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

Multiply fractions:

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

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

Cancel the common factor:

cos (α)

= \frac{1}{sin ( α )}

= \frac{1}{sin ( α )}

= \frac{1}{sin ( α )}

Rewrite using trig identities

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

\frac{1}{\frac{1}{c s c ( α )}}

Simplify

\frac{1}{\frac{1}{c s c ( α )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ( α )}{1}

Apply rule

\frac{a}{1} = a

= csc (α)

csc (α)

csc (α)

We showed that the two sides could take the same form

\Rightarrow True

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

\frac{1}{sec ( a ) + tan ( a )} = sec (a) - tan (a)

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

sin^{3} (θ) + sin (θ) cos^{2} (θ) = sin (θ)

sec^{2} (θ) cot^{2} (θ) = csc^{2} (θ)