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Consider the LHS: (1-sinA+cosA)2
= [(1-sinA) + cosA]2
= (1-sinA)2 + cos2A + 2(1-sinA)cosA
[ ∵ (a + b)2 = a2 + b2 + 2ab]
= 1 + sin2A − 2sinA + cos2A + 2(1-sinA)cosA [ ∵ (a - b)2 = a2 + b2 - 2ab]
= 1 + (sin2A + cos2A) − 2sinA + 2(1-sinA)cosA
= 1 + 1 − 2sinA + 2(1-sinA)cosA
[∵ sin2A + cos2A =1]
= 2 − 2sinA + 2(1-sinA)cosA
= 2(1 − sinA) + 2(1-sinA)cosA
= 2(1 − sinA)(1 + cosA)
= RHS
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