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25 February, 07:10

Sin^4x+2cos^2x-cos^4=1

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  1. 25 February, 07:14
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    Proven. We are done proving this identity because we get a true statement, which is 1 = 1.

    Step-by-step explanation:

    Sin^4x+2cos^2x-cos^4=1

    ...

    We want to prove : (Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1

    (Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1

    Factor this trinomial. considering ((cos x) ^2) is the variable

    so

    (((sin x) ^2) ^2 - ((cos x) ^2) ^2 + 2 (cos x) ^2 + 1 - 1 = 1

    (((sin x) ^2) ^2 - ((cos x) ^2) ^2 + 2 (cos x) ^2 - 1 + 1 = 1

    (((sin x) ^2) ^2 + [ - ((cos x) ^2) ^2 + 2 (cos x) ^2 - 1 ] + 1 = 1

    (((sin x) ^2) ^2 - [ ((cos x) ^2) - 1 ]^2 + 1 = 1

    But also notice that (sin x) ^2 = 1 - (cos x) ^2 from the trig identity:

    (sin x) ^2 + (cos x) ^2 = 1

    ((1 - (cos x) ^2) ^2 - [ ((cos x) ^2) - 1 ]^2 + 1 = 1

    here we see that (1 - (cos x) ^2) ^2 = [ ((cos x) ^2) - 1 ]^2

    so we get (0 + 1) = 1

    1 = 1 true.

    Proven. We are done proving this identity because we get a true statement.
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