Question

(cosa/2 +sina/2)^2 = 1+sina

Answer 1

Consider the LHS: (1-sinA+cosA)2 = [(1-sinA) + cosA]2
= (1-sinA)2 + cos2A + 2(1-sinA)cosA
= 1 + sin2A − 2sinA + cos2A + 2(1-sinA)cosA
= 1 + (sin2A + cos2A) − 2sinA + 2(1-sinA)cosA
= 1 + 1 − 2sinA + 2(1-sinA)cosA [Since, sin2A + cos2A =1]
= 2 − 2sinA + 2(1-sinA)cosA
= 2(1 − sinA) + 2(1-sinA)cosA
= 2(1 − sinA)(1 + cosA)
= RHS

Answer 2

$( \cos \frac{a}{2} + \sin \frac{a}{2}) {}^{2} \\ = { \cos}^{2} \frac{a}{2} + \sin {}^{2} \frac{a}{2} + 2 \sin \frac{a}{2} \: \cos \frac{a}{2} \\ = 1 + \sin \: a$
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