If `u=cot^(-1)sqrt(cos theta) -tan^(-1)sqrt(cos theta)` then sin u=

Options

A. `sin^(2) theta`
B. `cos^(2) theta`
C. `tan^(2) theta `
D. `tan^(2) 2theta`

Answer

C

Explanation

We have
`u = cot^(-1) sqrt(cos2 theta)-tan^(-1) sqrt(cos2 theta)`
`rarr u = (pi)/(2) - tan^(-1)sqrt(cos 2 theta)-tan^(-1)sqrt(cos 2 theta)`
`rarr u = (pi)/(2) -tan^(-1)sqrt(cos 2 theta)/(sin^(2)theta)` ltrbgt `rarr tan(pi)/(2)-u=sqrt(cos 2 theta)/(sin^(2)theta)`
`rarr cot u =sqrt(cos^(2) theta)/(sin^(2)theta)`
`rarr sin u = (sin^(2) theta)/sqrt(sin^(4)theta + 1 -2 sin^(2) theta)`
`rarr sin u =(sin^(2)theta)/(|1-sin^(2)theta|)=tan^(2)theta`