Question

Cosacos a/2+ sinasin a/2

Answer 1

$\cos( \alpha ) \times \cos( \frac{ \alpha }{2} ) + \sin( \alpha ) \times \sin( \frac{ \alpha }{2} ) = \cos( \alpha - \frac{ \alpha }{2} ) = \cos( \frac{1}{2} \alpha )$

Answer 2

It seems like you are trying to write a trigonometric expression, but there appears to be some confusion in the way you've written it. Let's clarify and solve the expression step by step.

If the expression you provided is:

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cos(a/2) + sin(a) * sin(a/2)

Then, we can simplify it using trigonometric identities. One useful identity is the double-angle identity for sine:

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sin(2θ) = 2sin(θ)cos(θ)

Now, let's focus on the given expression:

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cos(a/2) + sin(a) * sin(a/2)

We can use the double-angle identity for sine by setting θ = a/2:

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sin(a) = 2sin(a/2)cos(a/2)

Substitute this back into the original expression:

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cos(a/2) + 2sin(a/2)cos(a/2) * sin(a/2)

Now, we can factor out the common term of sin(a/2):

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cos(a/2) + 2sin^2(a/2)cos(a/2)

Next, we can use the Pythagorean identity for sine and cosine:

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sin^2(θ) + cos^2(θ) = 1

So, 2sin^2(a/2) can be replaced with 2(1 - cos^2(a/2)):

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cos(a/2) + 2(1 - cos^2(a/2)) * cos(a/2)

Expand the expression:

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cos(a/2) + 2cos(a/2) - 2cos^3(a/2)

Combine like terms:

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3cos(a/2) - 2cos^3(a/2)

That's the simplified expression: 3cos(a/2) - 2cos^3(a/2).