Question

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

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Solution

Let us consider two right angle triangles right angled at Q and S respectively.
Now,

cosA space equals space AS over AR

cosB space equals space BQ over BP

But cos A = cos B (given)
∴

AS over AR equals BQ over BP

rightwards double arrow

AS over BQ space equals space AR over BP

Let

AS over BQ equals AR over BP equals straight K

In ΔRAS,
Using Pythagoras theorem, we have

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In APBQ,
Using Pythagoras theorem, we have

PQ space equals space square root of BP squared minus BQ squared end root

So,

RS over PQ equals fraction numerator straight K. square root of BP squared minus BQ squared end root over denominator square root of BP squared minus BQ squared end root end fraction equals straight K space space... left parenthesis ii right parenthesis

Comparing (i) and (ii), we get

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So, by using SSS similar condition

space increment RSA space tilde space increment PQB

∴

angle straight A space equals space angle straight B.

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