Is cos(θ)=2cos^2(θ/2)-1 ?

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

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prove cos(θ)=2cos^2(θ/2)-1

Solution

prove

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Solution

True

Solution steps

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Let:

u = \frac{θ}{2}

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

Prove

cos (2 u) = 2 cos^{2} (u) - 1 :

True

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

Manipulating left side

cos (2 u)

Rewrite using trig identities

cos (2 u)

Rewrite as

= cos (u + u)

Use the Angle Sum identity:

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

= cos (u) cos (u) - sin (u) sin (u)

Simplify

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

cos (u) cos (u) - sin (u) sin (u)

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

cos (u) cos (u)

Apply exponent rule:

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

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

= cos^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= cos^{2} (u)

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

sin (u) sin (u)

Apply exponent rule:

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

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

= sin^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= sin^{2} (u)

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

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

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

Therefore

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos(θ)=2cos^2(θ/2)-1 ?
The answer to whether cos(θ)=2cos^2(θ/2)-1 is True