510४7 प8 (८050: + ८0500 + (डय + sinf)? = 4cos

1.	510४7 प8 (८050: + ८0500 + (डय + sinf)? = 4cos
Answer» => (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(cosAcosB - sinAsinB) => 1 + 1 + 2(cosAcosB - sinAsinB) => 2 + 2(cosAcosB - sinAsinB) => 2 (1 + (cosAcosB + sinAsinB)) => 2 * (1 + cos(A-B)) {Because: cosAcosB - sinAsinB = cos(A+B)} => 2 * 2cos^2 ((A+B)/2) => 4cos^2 (A+B)/2

Answer»

=> (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

Discussion