Question

Prove that (cosA+cosB)²+(sinA-sinB)²=4cos²(A/2+B/2)

Answer 1

Prove (cosA+cosB)^2+(sinA-sinB)^2 = 4cos^2(A+B)/2

(cosA+cosB)^2 + (sinA-sinB)^2

=> (cos^2A + cos^2B + 2cosAcosB) + (sin^2A + sin^2B - 2sinAsinB)

=> cos^2A + cos^2B + sin^2A + sin^2B + 2cosAcosB - 2sinAsinB

=> cos^2A + sin^2A + cos^2B + sin^2B + 2(cosA*cosB - sinA*sinB)

=> 1 + 1 + 2(cosA*cosB - sinA*sinB)

=> 2 + 2(cosA*cosB - sinA*sinB)

=> 2 (1 + (cosA*cosB + sinA*sinB))

=> 2 * (1 + cos(A-B))

{Because: cosA*cosB - sinA*sinB = cos(A+B)}

=> 2 * 2cos^2 ((A+B)/2)

=> 4cos^2 (A+B)/2




Hope it helps please mark me as brainliest