Is cos(7x)+cos(3x)=2cos(5x)cos(2x) ?

The answer to whether cos(7x)+cos(3x)=2cos(5x)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 cos(7x)+cos(3x)=2cos(5x)cos(2x)

prove

cos (7 x) + cos (3 x) = 2 cos (5 x) cos (2 x)

True

Solution steps

cos (7 x) + cos (3 x) = 2 cos (5 x) cos (2 x)

Manipulating left side

cos (7 x) + cos (3 x)

Rewrite using trig identities

cos (7 x) + cos (3 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{7 x + 3 x}{2}) cos (\frac{7 x - 3 x}{2})

Simplify

2 cos (\frac{7 x + 3 x}{2}) cos (\frac{7 x - 3 x}{2}) : 2 cos (5 x) cos (2 x)

2 cos (\frac{7 x + 3 x}{2}) cos (\frac{7 x - 3 x}{2})

\frac{7 x + 3 x}{2} = 5 x

\frac{7 x + 3 x}{2}

Add similar elements:

7 x + 3 x = 10 x

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

= 2 cos (5 x) cos (\frac{7 x - 3 x}{2})

\frac{7 x - 3 x}{2} = 2 x

\frac{7 x - 3 x}{2}

Add similar elements:

7 x - 3 x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= 2 cos (5 x) cos (2 x)

= 2 cos (5 x) cos (2 x)

= 2 cos (5 x) cos (2 x)

We showed that the two sides could take the same form

\Rightarrow True

cot^{4} (x) - csc^{4} (x) = - 2 cot^{2} (x) - 1

cot (- α) cos (- α) + sin (- α) = - csc (α)

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

sin (θ) cos (θ) tan (θ) = 1 - cos^{2} (θ)

cot^{2} (- θ) + 1 = csc^{2} (θ)

Is cos(7x)+cos(3x)=2cos(5x)cos(2x) ?
The answer to whether cos(7x)+cos(3x)=2cos(5x)cos(2x) is True