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)/(|d|)sqrt(d^(2)-a^(2))`
D. none of these

Answer

A

Explanation

We have,
`a sinx+bcos(x+theta)+b cos(x-theta)=d` for some real x
`impliesa sinx+2bcos x costheta=d`
`implies|d|lesqrt(a^(2)+4b^(2)cos^(2)theta)`
`impliesd^(2)lea^(2)+4b^(2)cos^(2)theta`
`implies(d^(2)-a^(2))/(4b^(2))lecos^(2)thetaimpliescosthetage""(1)/(2|b|)sqrt(d^(2)-a^(2))`
Hence, the minimum value of `cos thetais (1)/(1|b|)sqrt(d^(2)-a^(2))`