If `sin(theta+alpha)=a,cos^(2)(alpha+beta)=b,` then` sin(alpha-beta)=`

Options

A. `ab-(a-a^(2))(1-b^(2))`
B. `ab-sqrt((1-a)^(2)(1-b))`
C. `+-{asqrtb-sqrt((1-a^(2))(1-b))}`
D. `bsqrta-sqrt((1-a)^(2)(1-b))`

Answer

C

Explanation

We have,
`sin(theta+alpha)=a andcos(theta+beta)=pmsqrtb.`
`thereforecos(theta+alpha)=a andcos(theta+beta)=pmsqrt(1-b)`
Now, `sin(alpha-beta)=sin{theta+alpha)+(theta+beta)}`
`impliessin(alpha-beta)=sin(theta+alpha)cos(theta+beta)-cos(theta+alpha)sin(theta+beta)`
`impliessin (alpha-beta)=pmasqrtb-{pmsqrt(1-a^(2))}xx{pmsqrt(1-b)}`
`impliessin(alpha-beta)=pm{asqrtb-sqrt(1-a^(2))sqrt(1-b)}`