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प्रश्न
If `(1 + z)/(1 - z)` = cos 2θ + i sin 2θ, show that z = i tan θ
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उत्तर
`(1 + z)/(1 - z) = (cos 2theta + "i" sin 2theta)/1` .......`["Use" ("Nr" - "Dr")/("Nr" + "Dr") "componendo and dividendo rule"]`
`(1 + z - 1 + z)/(1 + z + 1 - z) = (cos 2theta + "i"sin2theta - 1)/(cos 2theta + "i"sintheta + 1)`
`(2z)/2 = (1 - 2sin^2theta + 2"i"sintheta costheta - 1)/(2cos^2theta - 1 + "i"sintheta costheta + 1)` .....[(x + iy) = I(y – xi)]
z = `(2sintheta["i"costheta - sintheta])/(2costheta[costheta + "i"sintheta])`
= `("i"sintheta[costheta + "i"sin theta])/(costheta[costheta + "i"sintheta])`
z = i tan θ
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