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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Simplify

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

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

Cancel the common factor:

sin^{2} (x)

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

Apply rule

\frac{a}{1} = a

= cos (2 x)

= cos (2 x)

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove cos^2(x)-sin^2(y)=1-2sin^2(y)prove

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

prove sin(x)+(cos^2(x))/(sin(x))=csc(x)prove

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

prove (cos(x)+1)/(sin(x)+tan(x))=cot(x)prove

\frac{cos ( x ) + 1}{sin ( x ) + tan ( x )} = cot (x)

prove sin(3x)+sin(x)=2sin(2x)cos(x)prove

sin (3 x) + sin (x) = 2 sin (2 x) cos (x)

prove cos(A+B)=cos(A)cos(B)-sin(A)sin(B)prove

cos (A + B) = cos (A) cos (B) - sin (A) sin (B)

Frequently Asked Questions (FAQ)

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