Question

If A + B + C = π, then prove that $cos^{2}\frac{A}{2} + cos^{2}\frac{B}{2} - cos^{2}\frac{C}{2} = 2 cos\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}$.

Answer 1

Answer:

Step-by-step explanation:

if A + B + C = π

then to prove

$cos^{2}\frac{A}{2} + cos^{2}\frac{B}{2} - cos^{2}\frac{C}{2} = 2 cos\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}$

let us take LHS

$=cos^{2}\frac{A}{2} + cos^{2}\frac{B}{2} - cos^{2}\frac{C}{2}$

as we know from half angle formula

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

so put all these values

$=\frac{1}{2} (1+cos\:A)+\frac{1}{2} (1+cos\:B)-\frac{1}{2} (1+cos\:C)\\\\=\frac{1}{2}+\frac{1}{2}(cosA+cosB-cosC)\\\\=\frac{1}{2}+\frac{1}{2}[(4cos\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2})-1]\\\\\\=2cos\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}$

=RHS

hence proved