If `a sin x+b cos(x+theta)+b cos(x-theta)=d,` then the minimum value of `|cos theta|` is equal to

Options

A. `(1)/(2|b|)sqrt(d^(2)-a^(2))`
B. `(1)/(2|a|)sqrt(d^(2)-a^(2))`
C. `(1)/(2|d|)sqrt(d^(2)-a^(2))`
D. none of these.

Answer

A

Explanation

`alpha sin x+b cos(x+theta)+b cos(x-theta)=d`
`rArr a sin x+2bcos x cos theta=d`
`rArr |d|lesqrt(a^(2)+4b^(2)cos^(2)theta)`
`rArr (d^(2)-alpha^(2))/(4 b^(2))lecos^(2)theta`
`rArr|cos theta|ge(sqrt(d^(2)-a^(2)))/(2|b|)`