(x sin a + y cos a)^2+ (x cos a – y sin a)^2 = x^2 + y^2

Question

(x sin a + y cos a)^2+ (x cos a – y sin a)^2 = x^2 + y^2

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Thanh Hà 6 months 2021-08-14T05:10:21+00:00 1 Answers 2 views 0

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    2021-08-14T05:11:23+00:00

    Answer:

    The identity sin (x + y) – sin (x – y) = 2 cos x sin y is verified

    Step-by-step explanation:

    Let us revise com rules in trigonometry

    sin(Ф + α) = sinФ cosα + cosФ sin α

    sin(Ф – α) = sinФ cos α – cosФ sinα

    To verify the identity sin (x + y) – sin (x – y) = 2 cos x sin y, take the left hand side and simplify it to give the right hand side

    ∵ L.H.S = sin (x + y) – sin (x – y)

    ∵ sin (x + y) = sin x cos y + cos x sin y

    ∵ sin (x – y) = sin x cos y – cos x sin y

    – Substitute then in the left hand side

    ∴ L.H.S = [sin x cos y + cos x sin y] – [sin x cos y – cos x sin y]

    – simplify it and remember (-)(-) = (+)

    ∴ L.H.S = sin x cos y + cos x sin y – sin x cos y + cos x sin y

    – Add the like terms

    ∵ sin x cos y – sin x cos y = 0

    ∵ cos x sin y + cos x sin y = 2 cos x sin y

    ∴ L.H.S = 2 cos x sin y

    ∵ R.H.S = 2 cos x sin y

    ∴ L.H.S = R.H.S

    The identity sin (x + y) – sin (x – y) = 2 cos x sin y is verified

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