(i)prove the identity cosx/tax(1+sinx)=1/sin x - 1. (ii) hence solve the equation cosx/tanx(1+sinx)=2, for 0 is less than equals to x is less than equals to 360.

Answer 1

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Answer 2

(i)cosx/[tanx(1 + sinx)] = cos²x/[(1 + sinx)sinx] = (1 - sin²x)/[sinx(1 + sinx)] = (1 - sinx)/sinx = 1/sinx - 1;

(ii) cosx/[tanx(1 + sinx)] = 2; 1/sinx - 1 = 2; 1/sinx = 3; sinx = 1/3; x = sin^-1(1/3) or x = π - sin^-1(1/3)