Is cos(2x)=2cos^2(x)-1 ?

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

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

prove

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

True

Solution steps

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

Manipulating left side

cos (2 x)

Rewrite using trig identities

cos (2 x)

Rewrite as

= cos (x + x)

Use the Angle Sum identity:

cos (s + s) = cos (s) cos (s) - sin (s) sin (s)

= cos (x) cos (x) - sin (x) sin (x)

Simplify

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

cos (x) cos (x) - sin (x) sin (x)

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

cos (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

Use the Pythagorean identity:

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

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

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

Simplify

cos^{2} (x) - (1 - cos^{2} (x)) : 2 cos^{2} (x) - 1

cos^{2} (x) - (1 - cos^{2} (x))

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

- (1 - cos^{2} (x))

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

Simplify

cos^{2} (x) - 1 + cos^{2} (x) : 2 cos^{2} (x) - 1

cos^{2} (x) - 1 + cos^{2} (x)

Group like terms

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

Add similar elements:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True