Question

Answer to the (i) question please​

Answer 1

Step-by-step explanation:

We have,

$A + B + C = \frac{3\pi}{2}....[1]$

Now,

$\cos^{2}(A) + \cos^{2}(B) - \cos^{2}(C)$

$= \frac{1 + \cos(2A) }{2}+ \frac{1 + \cos(2B)}{2} - \frac{1 + \cos(2C)}{2}$

$= \frac{1}{2} + \frac{1}{2} \{\cos(2A) + \cos(2B) - \cos(2C) \}$

$= \frac{1}{2} + \frac{1}{2} \{2\cos(A + B) \cos(A - B) - \cos(3\pi -2(A + B)) \}$

$= \frac{1}{2} + \frac{1}{2} \{2\cos(A + B) \cos(A - B) + \cos2(A + B)\}$

$= \frac{1}{2} + \frac{1}{2} \{2\cos(A + B) \cos(A - B) + 2\cos^{2} (A + B) - 1\}$

$= \frac{1}{2} + \frac{1}{2} \{2\cos(A + B) \cos(A - B) + 2\cos^{2} (A + B) \} - \frac{1}{2}$

$= \frac{1}{2} [ 2\cos(A + B) \{\cos(A - B) + \cos (A + B) \}]$

$= \cos( \frac{3\pi}{2} - C) . 2\cos(A) \cos (B)$

$= - 2 \cos(A) \cos (B) \cos( C)$