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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

cos (2 A)

Rewrite using trig identities

cos (2 A)

Rewrite as

= cos (A + A)

Use the Angle Sum identity:

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

= cos (A) cos (A) - sin (A) sin (A)

Simplify

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

cos (A) cos (A) - sin (A) sin (A)

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

cos (A) cos (A)

Apply exponent rule:

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

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

= cos^{1 + 1} (A)

Add the numbers:

1 + 1 = 2

= cos^{2} (A)

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

sin (A) sin (A)

Apply exponent rule:

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

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

= sin^{1 + 1} (A)

Add the numbers:

1 + 1 = 2

= sin^{2} (A)

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

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

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

2 sin^{2} (3 x) = 1 - cos (6 x)

sec^{2} (x) - 1 = \frac{tan ( x )}{cot ( x )}

sec (θ) cos (θ) = 1

cot (θ) (cot (θ) + tan (θ)) = csc^{2} (θ)

cos (\frac{π}{2} + y) = - sin (y)