Lets first deal with left hand side:

sin^2 x * tanx + cos^2 x * cot x + 2 sinx cos x = (sin^3 x)/cos x + (cos^3 x)/sinx + 2 sin x cos x

(simplifying substitute, tan x = sinx/cosx and cot x = cosx/sinx)

= [sin^4x+ cos^4 x+ 2sin^2x cos^2 x]/(sinx cosx) = [sin^2 x + cos^2 x]^2/(sinx cosx) = 1/(sinx cosx)

(using sin^2 x + cos^2 x =1)

Right Hand side: tanx + cot x = sinx/cosx + cosx/sinx = (sin^2x + cos^x)/(sinx cosx) = 1/(sinx cosx)

Since LHS= RHS. hence proved.

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