Differentiate in two ways, using product rule and otherwise, the function `(1+2tanx)(5+4cos x)` .Verify that the answer are the same.

Explanation

Let `u=1+2 \tan x, v=5+4 \cos x`


Using product rule:

`\frac{d}{dx}(uv)=uv'+u'v `

`\frac{d}{dx}(1+2tanx)(5+4cosx) `

`=(1+2tan x)(-4sin x)+(5+4cosx)(2sec^2x)`

`=-4sin x-8tanx sin x+10sec^2x+8sec x `

`=-4sin x+10sec^2 x (\frac{8}{cos x}-\frac{8 \sin^2x}{\cos x})`

`=-4sin x+10sec^2 x+8(\frac{\cos^2x}{\cos x})`

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