Is (csc^2(θ))/(cot(θ))=csc(θ)sec(θ) ?

The answer to whether (csc^2(θ))/(cot(θ))=csc(θ)sec(θ) is True

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prove (csc^2(θ))/(cot(θ))=csc(θ)sec(θ)

Solution

prove

\frac{csc ^{2} ( θ )}{cot ( θ )} = csc (θ) sec (θ)

Solution

True

Solution steps

\frac{csc ^{2} ( θ )}{cot ( θ )} = csc (θ) sec (θ)

Manipulating left side

\frac{csc ^{2} ( θ )}{cot ( θ )}

Express with sin, cos

\frac{csc ^{2} ( θ )}{cot ( θ )}

Use the basic trigonometric identity:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2}}{cot ( θ )}

Use the basic trigonometric identity:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2}}{\frac{c o s ( θ )}{s i n ( θ )}}

Simplify

\frac{( \frac{1}{s i n ( θ )} ) ^{2}}{\frac{c o s ( θ )}{s i n ( θ )}} : \frac{1}{sin ( θ ) cos ( θ )}

\frac{( \frac{1}{s i n ( θ )} ) ^{2}}{\frac{c o s ( θ )}{s i n ( θ )}}

Apply the fraction rule:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2} sin ( θ )}{cos ( θ )}

(\frac{1}{sin ( θ )})^{2} = \frac{1}{sin ^{2} ( θ )}

(\frac{1}{sin ( θ )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

= \frac{\frac{1}{s i n ^{2} ( θ )} sin ( θ )}{cos ( θ )}

Multiply

\frac{1}{sin ^{2} ( θ )} sin (θ) : \frac{1}{sin ( θ )}

\frac{1}{sin ^{2} ( θ )} sin (θ)

Multiply fractions:

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

= \frac{1 \cdot sin ( θ )}{sin ^{2} ( θ )}

Multiply:

1 \cdot sin (θ) = sin (θ)

= \frac{sin ( θ )}{sin ^{2} ( θ )}

Cancel the common factor:

sin (θ)

= \frac{1}{sin ( θ )}

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

Apply the fraction rule:

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

= \frac{1}{sin ( θ ) cos ( θ )}

= \frac{1}{sin ( θ ) cos ( θ )}

= \frac{1}{cos ( θ ) sin ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{cos ( θ ) \frac{1}{c s c ( θ )}}

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( θ )} \cdot \frac{1}{c s c ( θ )}}

Simplify

\frac{1}{\frac{1}{s e c ( θ )} \cdot \frac{1}{c s c ( θ )}}

Multiply

\frac{1}{sec ( θ )} \cdot \frac{1}{csc ( θ )} : \frac{1}{sec ( θ ) csc ( θ )}

\frac{1}{sec ( θ )} \cdot \frac{1}{csc ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 1}{sec ( θ ) csc ( θ )}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{sec ( θ ) csc ( θ )}

= \frac{1}{\frac{1}{s e c ( θ ) c s c ( θ )}}

Apply the fraction rule:

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

= \frac{sec ( θ ) csc ( θ )}{1}

Apply rule

\frac{a}{1} = a

= sec (θ) csc (θ)

sec (θ) csc (θ)

sec (θ) csc (θ)

= csc (θ) sec (θ)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (csc^2(θ))/(cot(θ))=csc(θ)sec(θ) ?
The answer to whether (csc^2(θ))/(cot(θ))=csc(θ)sec(θ) is True