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

sin^{4} (x) - 2 sin^{2} (x) + 1 = cos^{4} (x)

1 + sec (x) = \frac{cos ( x ) + 1}{cos ( x )}

tan (θ) + cot (θ) = csc (θ) sec (θ)

2 sin (x) + 1 = 3 csc (x)

sec (x) cot (x) = \frac{1}{sin ( x )}