Is (sin(3x)-sin(5x))/(2cos(4x))=-sin(x) ?

The answer to whether (sin(3x)-sin(5x))/(2cos(4x))=-sin(x) is True

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prove (sin(3x)-sin(5x))/(2cos(4x))=-sin(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Simplify

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

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

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

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

Add similar elements:

3 x + 5 x = 8 x

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Use the negative angle identity:

sin (- x) = - sin (x)

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

Remove parentheses:

(- a) = - a

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{8}{2} = 4

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

cos (4 x)

= - sin (x)

= - sin (x)

= - sin (x)

We showed that the two sides could take the same form

\Rightarrow True

Is (sin(3x)-sin(5x))/(2cos(4x))=-sin(x) ?
The answer to whether (sin(3x)-sin(5x))/(2cos(4x))=-sin(x) is True