If `tan 6 theta=p//q` find the value of `(1)/(2)(p co sec 2 theta-q sec 2 theta)` terms of p and q.

Explanation

Here, we have `tan 6 theta=p//q`
or `(sin 6 theta)/(cos6 theta)=(p)/(q)`
or `(p)/(sin 6theta)=(q)/(cos 6 theta)=(sqrt(p^(2)+q^(2)))/(sqrt(3))=sqrt(q^(2)+q^(2))=k`(say)
Now `y=(1)/(2)(p co sec 2 theta-q sec 2theta)`
`=(1)/(2)((p)/(sin 2theta)-(q)/(cos 2 theta))`
`=(1)/(2)[(p cos 2 theta-q sin 2theta)/(sin 2theta cos 2 theta)]`
`=[(2k sin 6 theta cos 2 theta-2k cos 6 theta sin 2 theta)/(4 sin 2 theta cos 2 theta)]`
`=k(sin(6theta-2theta))/(sin 4 theta)`
`=k=sqrt(p^(2)+q^(2))`