Question

(CosA-CosB)²+(SinA-SinB)²=4Sin²(A-B)/2​

Answer 1

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Answer 2

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

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

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)

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

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)

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)

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)

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))

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))

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)}

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)

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