If angle A and angle B are acute angles such that cos A = cos B, then show that angle A = angle B.

Given that: cos A= cos B
cos A = cos B
⇒ AP/AQ = BC/BD
⇒ AP/BC = AQ/BD
Let AP/BC = AQ/BC = k …(i)
Therefore, AP = k(BC) and AQ = k(BD)
Now, In △APQ and △BCD
PQ/CD = √((AQ²-AP²))/(√(BD²-BC²))
= (√((k.BD)²-(k.BC)²))/(√(BD²-BC²))
= (k√(BD²-BC²))/(√(BD²-BC²)) = k …(ii)
From the equation (i) and (ii), we get
AP/BC = AQ/BD = PQ/CD
So, △APQ ∼ △BCD [SSS similarity criteria]
Hence, ∠A = ∠B

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