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Sum of Trigonometric Ratios in Terms of Their Product

lf Sin2A+Co...

Question

lf $S i n^{2} A + C o s^{2} B + S i n^{2} C = 1$ , then the triangle $A B C$ is

A

isosceles

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B

equilateral

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C

right angled

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D

scalene

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Solution

The correct option is C right angled
$s i n^{2} A + c o s^{2} B = c o s^{2} C$
$s i n^{2} A = (c o s C - c o s B) (c o s C + c o s B)$
$s i n^{2} A = 2 \times 2 c o s (\frac{C + B}{2}) c o s \frac{(C - B)}{2} s i n (\frac{B + C}{2}) s i n (\frac{B - C}{2})$
$s i n^{2} A = s i n (B - C) s i n (B + C)$
$s i n A = s i n (B - C)$
$A = B - C$
$A + B + C = π$
$2 B = π$
$B = \frac{π}{2}$

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