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prove that sintheta.sec3theta+sin3theta....

prove that `sintheta.sec3theta+sin3theta.sec3^2theta+sin3^2theta.sec3^3theta+ u p ton` terms `=1/2[tan3^ntheta-tantheta]`

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`sin theta. Sec 3 theta+sin 3 theta. Sec 3^(2)theta.sec 3^(2)theta+sin3^(2)theta.sec3^(3)theta+`.......
upto n terms
`=sum_(r=1)^(n)sin3^(r-1)theta.sec3^(r)theta`
`=sum_(r=1)^(n)(2 cos 3 ^(r-1)thetasin 3^(r-1)theta)/(cos3^(r01)theta.cos 3^(r)theta)`
`=(1)/(2)sum_(r=1)^(n)(sin(2.3^(r-1)theta))/(cos3^(r-1)theta. cos3^(r)theta)`
`=(1)/(2)sum_(r=1)^(n)(sin(3^(r)theta-3^(r-1)theta))/(cos 3^(r-1)theta.cos 3^(r)theta)`
`(1)/(2)sum_(r=1)^(n)(sin3^(r)theta. cos3^(r-1)theta-cos3^(r)theta. sin 3^(r-1)theta)/(cos3^(r-1)theta. cos 3^(r)theta)`
`=(1)/(2)sum_(r=1)^(n)(tan 3^(r)theta-tan3^(r-1)theta)`
`=(1)/(2)[tan 3^(n)theta-tan theta]`
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