Is (sin(3x)+sin(5x))/(cos(3x)-cos(5x))=cot(x) ?

The answer to whether (sin(3x)+sin(5x))/(cos(3x)-cos(5x))=cot(x) is True

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Popular Trigonometry >

prove (sin(3x)+sin(5x))/(cos(3x)-cos(5x))=cot(x)

Solution

prove

\frac{sin ( 3 x ) + sin ( 5 x )}{cos ( 3 x ) - cos ( 5 x )} = cot (x)

Solution

True

Solution steps

\frac{sin ( 3 x ) + sin ( 5 x )}{cos ( 3 x ) - cos ( 5 x )} = cot (x)

Manipulating left side

\frac{sin ( 3 x ) + sin ( 5 x )}{cos ( 3 x ) - cos ( 5 x )}

Rewrite using trig identities

\frac{sin ( 3 x ) + sin ( 5 x )}{cos ( 3 x ) - cos ( 5 x )}

Use the Sum to Product identity:

sin (s) + sin (t) = 2 sin (\frac{s + t}{2}) cos (\frac{s - t}{2})

= \frac{2 sin ( \frac{3 x + 5 x}{2} ) cos ( \frac{3 x - 5 x}{2} )}{cos ( 3 x ) - cos ( 5 x )}

Simplify

\frac{2 sin ( \frac{3 x + 5 x}{2} ) cos ( \frac{3 x - 5 x}{2} )}{cos ( 3 x ) - cos ( 5 x )} : \frac{2 cos ( x ) sin ( 4 x )}{cos ( 3 x ) - cos ( 5 x )}

\frac{2 sin ( \frac{3 x + 5 x}{2} ) cos ( \frac{3 x - 5 x}{2} )}{cos ( 3 x ) - cos ( 5 x )}

2 sin (\frac{3 x + 5 x}{2}) cos (\frac{3 x - 5 x}{2}) = 2 sin (\frac{8 x}{2}) cos (- \frac{2 x}{2})

2 sin (\frac{3 x + 5 x}{2}) cos (\frac{3 x - 5 x}{2})

Add similar elements:

3 x + 5 x = 8 x

= 2 sin (\frac{8 x}{2}) cos (\frac{3 x - 5 x}{2})

\frac{3 x - 5 x}{2} = - \frac{2 x}{2}

\frac{3 x - 5 x}{2}

Add similar elements:

3 x - 5 x = - 2 x

= \frac{- 2 x}{2}

Apply the fraction rule:

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

= - \frac{2 x}{2}

= 2 sin (\frac{8 x}{2}) cos (- \frac{2 x}{2})

= \frac{2 sin ( \frac{8 x}{2} ) cos ( - \frac{2 x}{2} )}{cos ( 3 x ) - cos ( 5 x )}

Use the negative angle identity:

cos (- x) = cos (x)

= \frac{2 cos ( \frac{2 x}{2} ) sin ( \frac{8 x}{2} )}{cos ( 3 x ) - cos ( 5 x )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{2 cos ( x ) sin ( \frac{8 x}{2} )}{cos ( 3 x ) - cos ( 5 x )}

Divide the numbers:

\frac{8}{2} = 4

= \frac{2 cos ( x ) sin ( 4 x )}{cos ( 3 x ) - cos ( 5 x )}

= \frac{2 cos ( x ) sin ( 4 x )}{cos ( 3 x ) - cos ( 5 x )}

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= \frac{2 cos ( x ) sin ( 4 x )}{- 2 sin ( \frac{3 x + 5 x}{2} ) sin ( \frac{3 x - 5 x}{2} )}

Simplify

\frac{2 cos ( x ) sin ( 4 x )}{- 2 sin ( \frac{3 x + 5 x}{2} ) sin ( \frac{3 x - 5 x}{2} )} : \frac{cos ( x )}{sin ( x )}

\frac{2 cos ( x ) sin ( 4 x )}{- 2 sin ( \frac{3 x + 5 x}{2} ) sin ( \frac{3 x - 5 x}{2} )}

- 2 sin (\frac{3 x + 5 x}{2}) sin (\frac{3 x - 5 x}{2}) = - 2 sin (\frac{8 x}{2}) sin (- \frac{2 x}{2})

- 2 sin (\frac{3 x + 5 x}{2}) sin (\frac{3 x - 5 x}{2})

Add similar elements:

3 x + 5 x = 8 x

= - 2 sin (\frac{8 x}{2}) sin (\frac{3 x - 5 x}{2})

\frac{3 x - 5 x}{2} = - \frac{2 x}{2}

\frac{3 x - 5 x}{2}

Add similar elements:

3 x - 5 x = - 2 x

= \frac{- 2 x}{2}

Apply the fraction rule:

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

= - \frac{2 x}{2}

= - 2 sin (\frac{8 x}{2}) sin (- \frac{2 x}{2})

= \frac{2 cos ( x ) sin ( 4 x )}{- 2 sin ( \frac{8 x}{2} ) sin ( - \frac{2 x}{2} )}

Apply the fraction rule:

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

= - \frac{2 cos ( x ) sin ( 4 x )}{2 sin ( \frac{8 x}{2} ) sin ( - \frac{2 x}{2} )}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{cos ( x ) sin ( 4 x )}{sin ( \frac{8 x}{2} ) sin ( - \frac{2 x}{2} )}

Use the negative angle identity:

sin (- x) = - sin (x)

= - \frac{cos ( x ) sin ( 4 x )}{( - sin ( \frac{2 x}{2} ) ) sin ( \frac{8 x}{2} )}

Refine

= \frac{cos ( x ) sin ( 4 x )}{sin ( \frac{2 x}{2} ) sin ( \frac{8 x}{2} )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{cos ( x ) sin ( 4 x )}{sin ( x ) sin ( \frac{8 x}{2} )}

Divide the numbers:

\frac{8}{2} = 4

= \frac{cos ( x ) sin ( 4 x )}{sin ( x ) sin ( 4 x )}

Cancel the common factor:

sin (4 x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (sin(3x)+sin(5x))/(cos(3x)-cos(5x))=cot(x) ?
The answer to whether (sin(3x)+sin(5x))/(cos(3x)-cos(5x))=cot(x) is True