Is cos(3x)+cos(x)=2cos(2x)cos(x) ?

The answer to whether cos(3x)+cos(x)=2cos(2x)cos(x) is True

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

prove

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

True

Solution steps

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

Manipulating left side

cos (3 x) + cos (x)

Rewrite using trig identities

cos (3 x) + cos (x)

Use the Sum to Product identity:

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

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

Simplify

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

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

\frac{3 x + x}{2} = 2 x

\frac{3 x + x}{2}

Add similar elements:

3 x + x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

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

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

\frac{3 x - x}{2}

Add similar elements:

3 x - x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 cos (2 x) cos (x)

= 2 cos (2 x) cos (x)

= 2 cos (2 x) cos (x)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e sin (4 x) = 2 sin (2 x) cos (2 x)

p ro v e \frac{cos ( x ) cot ( x )}{1 - sin ( x )} - 1 = csc (x)

p ro v e (1 + tan (x))^{2} = sec^{2} (x) + 2 tan (x)

p ro v e \frac{sec ^{2} ( x )}{2 tan ( x )} = csc (2 x)

p ro v e (sec^{2} (x) - 1) csc (x) = tan (x) sec (x)

Is cos(3x)+cos(x)=2cos(2x)cos(x) ?
The answer to whether cos(3x)+cos(x)=2cos(2x)cos(x) is True