of sinA=sinx +siny _________ 1- Sinx.siny then show that, COS A = +,- of cosx. cosy/

Question

of
sinA=sinx +siny
_________
1- Sinx.siny
then show that,

COS A = +,- of cosx. cosy/
___________
1+sinx.siny​

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Hadley 1 month 2021-08-12T17:14:40+00:00 1 Answer 1 views 0

Answers ( )

    0
    2021-08-12T17:15:45+00:00

    Given : sinA = (sinx + siny)/(1 + sinx siny)

    To prove : cosA = ± cosx cosy/(1 + sinx siny)

    proof : sinA = (sinx + siny)/(1 + sinx siny) …..(1)

    we know, sin²A + cos²A = 1

    so, cosA = ± √(1 – sin²A)

    from equation (1) we get,

    = ± √[1 – (sinx + siny)²/(1 + sinx siny)²]

    = ± √[{(1 + sinx siny)² – (sinx + siny)²}/(1 + sinx siny)²]

    = ± √[(1 + sin²x sin²y + 2sinx siny – sin²x – sin²y – 2sinx siny)/(1 + sinx siny)² ]

    = ± √(sin²x sin²y – sin²x – sin²y + 1)/(1 + sinx siny)

    = ± √{(-sin²x(1 – sin²y) + 1(1 – sin²y)}/(1 + sinx siny)

    = ± √(1 – sin²x)(1 – sin²y)/(1 + sinx siny)

    = ± √(cos²x cos²y)/(1 + sinx siny)

    = ± cosx cosy/(1 + sinx siny)

    Therefore cosA = ± cosx cosy/(1 + sinx siny)

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