Maths-
General
Easy

Question

4 sin to the power of 4 space x plus cos to the power of 4 space x equals 1 if

  1. x equals n pi
  2. straight x equals straight n pi plus-or-minus 1 half cos to the power of negative 1 end exponent space open parentheses 1 fifth close parentheses
  3. straight x equals fraction numerator straight n pi over denominator 2 end fraction
  4. text  none of these,  end text left parenthesis n element of I right parenthesis

hintHint:

In this question, we have given, 4 sin to the power of 4 space x plus cos to the power of 4 space x equals 1. Solve this equation and find the general solution for x.

The correct answer is: x equals n pi


    Here, we have to find the the general solution for x
    Firstly, we have given equation,
    4 sin to the power of 4 space x plus cos to the power of 4 space x equals 1

    equals greater than 4 s i n to the power of 4 x equals 1 minus c o s to the power of 4 x equals left parenthesis 1 minus c o s squared x right parenthesis left parenthesis 1 plus c o s squared x right parenthesis equals s i n squared x left parenthesis 2 minus s i n squared x right parenthesis
equals greater than 4 s i n to the power of 4 x equals s i n squared x left parenthesis 2 minus s i n squared x right parenthesis
equals greater than 4 s i n to the power of 4 x equals 2 s i n squared x minus s i n to the power of 4 x
equals greater than 5 s i n to the power of 4 x equals 2 s i n squared x
equals greater than 5 s i n to the power of 4 x minus 2 s i n squared x equals 0
equals greater than s i n squared x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket 5 s i n squared x minus 2 right square bracket equals 0

    So we can write,
    S i n squared x equals 0 space space a n d space space 5 s i n squared x minus 2 equals 0
    x = n π and sin2x = 2 over 5
    -2 sin2 x = fraction numerator negative 4 over denominator 5 end fraction
    1 – 2 sin2 x = 1 fifth
    Cos 2x = 1 fifth
    2x = 2n π ± cos-1 (1 fifth)
    X = n π ±1 half cos-1 (1 fifth)
    Therefore, the correct answer is x = n π and x = n π ±1 half cos-1 (1 fifth). Both options will correct.

    In this question, we have to find the general solution of x. Here more than one option will correct. Remember the rules for finding the general solution.

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