1.

The value of `(2sinx)/(sin3x)+(tanx)/(tan3x)`is________.

Answer» `(2sinx)/(sin3x) +tanx/(tan3x)`
`=(2sinx)/(3sinx-4sin^3x) +tanx/((3tanx - tan^3x)/(1-3tan^2x))`
`=2/(3-4sin^2x) +(1-3tan^2x)/(3-tan^2x)`
`=2/(3-4sin^2x) +(1-3sin^2x/cos^2x)/(3-sin^2x/cos^2x)`
`=2/(3-4sin^2x) +(1-3sin^2x/(1-sin^2x))/(3-sin^2x/(1-sin^2x))`
`=2/(3-4sin^2x) +(1-sin^2x - 3sin^2x)/(3-3sin^2x-sin^2x)`
`=2/(3-4sin^2x) +(1-4sin^2x)/(3-4sin^2x)`
`=(2+1-4sin^2x) /(3-4sin^2x) `
`=(3-4sin^2x) /(3-4sin^2x) `
`=1`
`:. (2sinx)/(sin3x) +tanx/(tan3x) = 1.`


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