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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

sec^{2} (x) - 1

Express with sin, cos

- 1 + sec^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{( \frac{1}{c s c ( x )} ) ^{2}}{cos ^{2} ( x )}

Use the basic trigonometric identity:

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

\frac{( \frac{1}{c s c ( x )} ) ^{2}}{( \frac{1}{s e c ( x )} ) ^{2}}

Simplify

\frac{( \frac{1}{c s c ( x )} ) ^{2}}{( \frac{1}{s e c ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{( \frac{1}{c s c ( x )} ) ^{2}}{\frac{1}{s e c ^{2} ( x )}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{c s c ^{2} ( x )}}{\frac{1}{s e c ^{2} ( x )}}

Divide fractions:

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

= \frac{1 \cdot sec ^{2} ( x )}{csc ^{2} ( x ) \cdot 1}

Refine

= \frac{sec ^{2} ( x )}{csc ^{2} ( x )}

\frac{sec ^{2} ( x )}{csc ^{2} ( x )}

\frac{sec ^{2} ( x )}{csc ^{2} ( x )}

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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