cos^2(45-a)-sin^2(45-a)=sin2a
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Using the addition identity for sine
sin(x + y) = sinxcosy - cosxsiny
Consider the left side
cos²(45 - A) - sin²(45 - A)
cos²(45 - A) = 1 - sin²(45 - A), thus
1 - sin²(45 - A) - sin²(45 - A)
= 1 - 2sin²(45 - A) ← expand sin(45 - A)
= 1 - 2(sin45cosA - cos45sinA)²
= 1 - 2(cosA - sinA)²
= 1 - 2(cos²A - sinAcosA + sin²A)
= 1 - cos²A + 2sinAcosA - sin²A
= sin²A + 2sinAcosA - sin²A
= 2sinAcosA
= sin2A = right side ⇒ verified
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