Question

if tan theta=4/3 show: root 1-sin theta/1+sin theta+ root 1+sin theta/1-sin theta=10/3

Answer 1

I don’t know the answer to this question

Answer 2

EXPLANATION:

tan theta = 4/3 = opposite side/adjacent side

By writing the given data on the opposite and adjacent side of a right-angled triangle,

By the Pythagorus theorem,

Hypotenuse =root (4^2+3^2)

sin theta = opposite side/hypotenuse = 4/5

cos theta = adjacent side/hypotenuse = 3/5

NOW,

L.H.S= root {1-sin theta}/{1+sin theta} + root{1+sin theta}/{1-sin theta}

       = root{(1-sin theta)(1-sin theta)} / {(1+sin theta)(1-sin theta)}  +                 root{(1+sin theta)(1+sin theta)} / {(1-sin theta)(1+sin theta)}

(Taking conjugate)

       =  root{(1-sin theta)^{2}} / {(1+sin theta)(1-sin theta)} +                 root{(1+sin theta)^{2}} / {(1-sin theta)(1+sin theta)}

       = root {(1-sin theta)^{2}} / {(1^{2}-sin^{2} theta)} + root{(1+sin theta)^{2}} / {(1^{2}-sin^{2} theta)}                    

            (1- sin^{2} theta = cos^{2} theta)

       = root{(1-sin theta)^{2}} / {(cos^{2} theta)} +  root{(1+sin theta)^{2}} / {(cos^{2} theta)}

       = {1- sin theta} / {cos theta} + {1+sin theta} / {cos theta}

       = {1- sin theta + 1 + sin theta} / {cos theta}                  

          (Taking L.C.M.)

       = {2} / {cos theta}

       = {2} / {3/5}

       = 10/3

       = R.H.S.

                                            HENCE PROVED.

Hope this helped..