(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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Ben Gia 6 months 2021-08-14T05:07:23+00:00 1 Answers 3 views 0

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    2021-08-14T05:08:34+00:00

    Answer:

    Proved

    Step-by-step explanation:

    Required

    Prove that:

    (x\ sin\ a + y\ cos\ a)^2 + (x\ cos\ a - y\ sin\ a)^2 = x^2 + y^2

    Solving from left to right:

    Open brackets

    (x\ sin\ a + y\ cos\ a)(x\ sin\ a + y\ cos\ a) + (x\ cos\ a - y\ sin\ a)(x\ cos\ a - y\ sin\ a) = x^2 + y^2x^2\ sin^2 a + 2xy\ sin\ a\ cos\ a + y^2\ cos^2 a + x^2\ cos^2 a - 2xy\ sin\ a\ cos\ a + y^2\ sin^2 a = x^2 + y^2

    Collect Like Terms

    x^2\ sin^2 a + 2xy\ sin\ a\ cos\ a  - 2xy\ sin\ a\ cos\ a + y^2\ cos^2 a + x^2\ cos^2 a+ y^2\ sin^2 a = x^2 + y^2

    x^2\ sin^2 a  + y^2\ cos^2 a + x^2\ cos^2 a+ y^2\ sin^2 a = x^2 + y^2

    Collect Like Terms

    x^2\ sin^2 a + y^2\ sin^2 a + x^2\ cos^2 a + y^2\ cos^2 a = x^2 + y^2

    Factorize:

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

    Further factorize

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

    In trigonometry:

    sin^2 a + cos^2 a = 1

    So, we have:

    (x^2 + y^2)(1) = x^2 + y^2

    x^2 + y^2 = x^2 + y^2

    Proved

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