Is (csc^2(θ)-1)/(cos(θ))=cot(θ)csc(θ) ?

The answer to whether (csc^2(θ)-1)/(cos(θ))=cot(θ)csc(θ) is True

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prove (csc^2(θ)-1)/(cos(θ))=cot(θ)csc(θ)

Solution

prove

\frac{csc ^{2} ( θ ) - 1}{cos ( θ )} = cot (θ) csc (θ)

Solution

True

Solution steps

\frac{csc ^{2} ( θ ) - 1}{cos ( θ )} = cot (θ) csc (θ)

Manipulating left side

\frac{csc ^{2} ( θ ) - 1}{cos ( θ )}

Express with sin, cos

\frac{- 1 + csc ^{2} ( θ )}{cos ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{- 1 + ( \frac{1}{s i n ( θ )} ) ^{2}}{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{- 1 + \frac{1}{s i n ^{2} ( θ )}}{cos ( θ )}

Join

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

- 1 + \frac{1}{sin ^{2} ( θ )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

Apply the fraction rule:

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

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

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

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

Rewrite using trig identities

\frac{1 - sin ^{2} ( θ )}{cos ( θ ) sin ^{2} ( θ )}

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

1 - sin^{2} (x) = cos^{2} (x)

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

Cancel the common factor:

cos (θ)

= \frac{cos ( θ )}{sin ^{2} ( θ )}

= \frac{cos ( θ )}{sin ^{2} ( θ )}

Manipulating right side

cot (θ) csc (θ)

Express with sin, cos

cot (θ) csc (θ)

Use the basic trigonometric identity:

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

= \frac{cos ( θ )}{sin ( θ )} csc (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

cos (θ) \cdot 1 = cos (θ)

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

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin^{2} (θ)

= \frac{cos ( θ )}{sin ^{2} ( θ )}

= \frac{cos ( θ )}{sin ^{2} ( θ )}

= \frac{cos ( θ )}{sin ^{2} ( 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)-1)/(cos(θ))=cot(θ)csc(θ) ?
The answer to whether (csc^2(θ)-1)/(cos(θ))=cot(θ)csc(θ) is True