Is (csc^2(x)-1)sec(x)=cot(x)csc(x) ?

The answer to whether (csc^2(x)-1)sec(x)=cot(x)csc(x) is True

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prove (csc^2(x)-1)sec(x)=cot(x)csc(x)

Solution

prove

(csc^{2} (x) - 1) sec (x) = cot (x) csc (x)

Solution

True

Solution steps

(csc^{2} (x) - 1) sec (x) = cot (x) csc (x)

Manipulating left side

(csc^{2} (x) - 1) sec (x)

Express with sin, cos

(- 1 + csc^{2} (x)) sec (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

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

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

Multiply:

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

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

Remove parentheses:

(- a) = - a

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

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

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

(\frac{1}{sin ( x )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Join

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

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

Convert element to fraction:

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

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

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

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

Apply the fraction rule:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

cos (x)

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

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

Manipulating right side

cot (x) csc (x)

Express with sin, cos

cot (x) csc (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

cos (x) \cdot 1 = cos (x)

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

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

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (csc^2(x)-1)sec(x)=cot(x)csc(x) ?
The answer to whether (csc^2(x)-1)sec(x)=cot(x)csc(x) is True