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

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

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Divide fractions:

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

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

Cancel the common factor:

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

cot (x) cot (x)

Simplify

cot (x) cot (x) : cot^{2} (x)

cot (x) cot (x)

Apply exponent rule:

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

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

= cot^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

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