(cosα+cosβ)^2+(sinα+sinβ)^2 is (a) 4 cos^2((α+β)/(2)) (b) 4 cos^2((α-β)/(2)) (c) -4 sin^2 (α)/(2

step 1 So in the given question we are told to simplify the expression cos alpha plus beta, cost of alp

(cosα+cosβ)^2+(sinα+sinβ)^2 is (a) 4 cos^2((α+β)/(2)) (b) 4...

$(\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}$ is
(a) $4 \cos ^{2}\left(\frac{\alpha+\beta}{2}\right)$
(b) $4 \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$
(c) $-4 \sin ^{2} \frac{\alpha}{2} \cos ^{2} \frac{\beta}{2}$
(d) $4 \sin \frac{\alpha}{2} \cos \frac{\beta}{2}$

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Key Concepts

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the square roots of expressions involving the cosine of the original angle. They are valuable for rewriting and simplifying expressions, particularly when dealing with squared trigonometric terms.

Sum-to-Product Identities

These identities convert sums of trigonometric functions into products, which can simplify complex expressions. They help in transforming the sum of two cosines or sines into a form that is easier to manipulate or integrate.

Pythagorean Identity

This identity states that for any angle, the sum of the squares of the sine and cosine equals one. It is fundamental in trigonometry and is often used to simplify expressions that involve both sine and cosine functions.

Cosine of Difference Identity

This identity relates the cosine of the difference of two angles to the product of the cosines of the individual angles plus the product of the sines of the individual angles. It's useful for combining or simplifying expressions where sine and cosine terms appear together.

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