Is 1/(1-tan^2(x))+1/(1-cot^2(x))=1 ?

The answer to whether 1/(1-tan^2(x))+1/(1-cot^2(x))=1 is True

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prove 1/(1-tan^2(x))+1/(1-cot^2(x))=1

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

\frac{1}{1 - ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} + \frac{1}{1 - ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

Join

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

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

Multiply:

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

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

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

Apply the fraction rule:

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

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

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

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

Apply exponent rule:

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

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

Join

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

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

Multiply:

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

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

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

Apply the fraction rule:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (x) + cos (x)) (sin (x) - cos (x))

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

Factor

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (cos (x) + sin (x)) (cos (x) - sin (x))

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

Least Common Multiplier of

(sin (x) + cos (x)) (sin (x) - cos (x)), (cos (x) + sin (x)) (cos (x) - sin (x)) : - (sin (x) + cos (x)) (sin (x) - cos (x))

(sin (x) + cos (x)) (sin (x) - cos (x)), (cos (x) + sin (x)) (cos (x) - sin (x))

Lowest Common Multiplier (LCM)

cos (x) - sin (x) = - (sin (x) - cos (x))

= (sin (x) + cos (x)) (sin (x) - cos (x)), - (cos (x) + sin (x)) (sin (x) - cos (x))

Compute an expression comprised of factors that appear either in

(sin (x) + cos (x)) (sin (x) - cos (x))

(cos (x) + sin (x)) (cos (x) - sin (x))

= - (sin (x) + cos (x)) (sin (x) - cos (x))

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

- (sin (x) + cos (x)) (sin (x) - cos (x))

For

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

multiply the denominator and numerator by

- 1

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

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

Since the denominators are equal, combine the fractions:

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

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

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

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

Cancel

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

Cancel

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

\frac{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}{( sin ( x ) + cos ( x ) ) ( sin ( x ) - cos ( x ) )}

sin (x) - cos (x) = - (cos (x) - sin (x))

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

Refine

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

Cancel the common factor:

cos (x) + sin (x)

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

Cancel the common factor:

cos (x) - sin (x)

= - 1

= - 1

= - (- 1)

Apply rule

- (- a) = a

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is 1/(1-tan^2(x))+1/(1-cot^2(x))=1 ?
The answer to whether 1/(1-tan^2(x))+1/(1-cot^2(x))=1 is True