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3 October, 08:10

Prove that

Cos (A+B) + Sin (A-B) = 2sin (45+A). cos (45+B)

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  1. 3 October, 08:19
    0
    Step-by-step explanation:

    cos (A+B) + sin (A-B) = 2 sin (45°+A) cos (45° + B)

    = 2 (sin45°cosA + cos45°sinA) (cos45°cosB - sin45°sinB)

    But sin45=cos45 = (sqrt2) / 2

    = 2 ((sqrt2) / 2 * cosA + (sqrt2) / 2 * sinA) ((sqrt2) / 2 * cosB - (sqrt2) / 2 * sinB)

    = 2 ((sqrt2) / 2 * (cosA + sinA)) * ((sqrt2) / 2 * (cosB - sinB))

    = 2 * (sqrt2) / 2 * (sqrt2) / 2 * (cosA + sinA) * (cosB - sinB)

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

    = cosAcosB + sinAcosB - cosAsinB - sinAsinB

    Regrouping:

    = (cosAcosB - sinAsinB) + (sinAcosB - cosAsinB)

    = cos (A+B) + sin (A-B)
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