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If cos25+sin25=k then the value os cos20 is ?

  • math -

    cos20 = cos(70-50) = cos70cos50 + sin70sin50

    cos25+sin25 = √2 sin70 = k, so
    sin70 = k/√2
    cos70 = √(2-k^2) / √2

    cos^2 25 + 2sin25cos25 + sin^2 25 = k^2
    1 + sin50 = k^2
    sin50 = k^2-1
    cos50 = k√(2-k^2)

    so, now we have

    cos 20 = k√(2-k^2)√(2-k^2) / √2 + k(k^2-1)/√2
    = (k(2-k^2) + k(k^2-1))/√2
    = k/√2

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