In $△ A B C$ , the value of $c o s^{2} A + c o s^{2} B - c o s^{2} C + 2 s i n A s i n B s i n C = ?$

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Solution

The correct option is D 1
$c o s (2 A) + c o s (2 B) - c o s (2 C) = 2 c o s (A + B) c o s (A - B) + 1 - 2 c o s ² (C)$
$= 2 c o s (π - C) c o s (A - B) + 1 - 2 c o s ² (C) = - 2 c o s (C) c o s (A - B) + c o s (C) + 1$
$= - 4 c o s (C) c o s (A - B + C) / 2 c o s (A - B - C) / 2 + 1$
$(A - B + C) / 2 = π / 2 - B a n d (A - B - C) / 2 = A - π / 2$
$c o s (2 A) + c o s (2 B) - c o s (2 C) = 1 - 4 s i n (A) s i n (B) c o s (C)$

$=$ $\frac{1 + c o s 2 A}{2}$ + $\frac{1 + c o s 2 B}{2}$ - $\frac{1 + c o s 2 C}{2}$ $+ 2 s i n A s i n B s i n C$

Now
$c o s 2 A + c o s 2 B - c o s 2 C + 2 s i n A s i n B s i n C$
$=$ $\frac{c o s 2 A + c o s 2 B - c o s 2 C}{2}$ $+ 2 s i n A s i n B s i n C$ + $\frac{1}{2}$
put the value $c o s 2 A + c o s 2 B - c o s 2 C$
and you will get answer $1$

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