InterviewSolution
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InterviewSolution
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If `sin^4 alpha + 4 cos^4 beta + 2 = 4sqrt2 sin alpha cos beta ; alpha, beta in [0,pi],` then `cos(alpha+beta) - cos(alpha - beta)` is equal to :A. `-1`B. `sqrt(2)`C. `-sqrt(3)`D. 0 |
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Answer» Correct Answer - C By applying `AM ge GM` inequality, on the numbers `sin^(4) alpha, 4 cos^(2) beta`, 1 and 1, we get `(sin^(4) alpha + 4 cos^(2) beta + 2)/(4)le (( sin^(4) alpha )(4cos^(4) beta).1.)^(1//4)` `rArr sin^(4) alpha + 4 cos^(4) Beta + 2 ge 4sqrt(2)sin alpha cos beta` But, it is given that `sin^(4) alpha + 4 cos^(4) beta + 2 = 4sqrt(2)sin alpha cos beta` So, `sin^(4) alpha= 4 cos ^(4) beta =1` `[because In AM ge GM`, equality holds when all given positive quantites are equal.] `rArr sin alpha = 1` and `sin beta = (1)/(sqrt(2)) " "......(i)` Now, `cos (alpha + beta) -cos (alpha + beta)= - 2sin alpha sin beta " "[because alpha, beta in [0, pi]]` `[because cos C - cos D = 2sin .(C + D)/(2)sin .(D-C)/(2)]` `= -2 xx 1 xx (1)/(sqrt(2)) " "["From Eq. (i)"]` `= - sqrt(2)` |
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