Prove the following identities csc^2 x / cos^2 x = sec^2 x csc^2 x (sin x / 1-cotx) + cosx/1-tanx = sinx + cosx

llltkl | Student

1) L.H.S.=`(csc^2 x) / (cos^2 x)=(csc^2 x)*1 / (cos^2 x)`

Since, `1/cos^2 x=sec^2 x`   So,

L.H.S.=`sec^2 x* csc^2 x` =R.H.S.

Hence, the proof.

2) L.H.S.=`(sin x) / (1-cotx) + (cosx)/(1-tanx)`

`=sinx/ (1-cosx/sinx) + (cosx)/(1-sinx/cosx)`

`=sinx/ ((sinx-cosx)/sinx) + (cosx)/((cosx-sinx)/cosx)`

`=(sin^2x)/ (sinx-cosx) - (cos^2x)/(sinx-cosx)`


`=sinx+cosx` =R.H.S.

Hence, the proof.

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