Question

(4cosx+sin^2x)^2/Inx

Answer 1

The simplified expression is
`\((24\sin^2 x + \sin^4 x)/(\ln x)\)` for
`\(((4\cos x + \sin^2 x)^2)/(\ln x)\).`

To simplify the expression
`\(((4\cos x + \sin^2 x)^2)/(\ln x)\)`, we'll break it down step by step. First, expand the numerator:

`\[(4\cos x + \sin^2 x)^2 = 16\cos^2 x + 8\sin^2 x + \sin^4 x.\]`

Now, substitute this back into the original expression:

`\[(16\cos^2 x + 8\sin^2 x + \sin^4 x)/(\ln x).\]`

To further simplify, factor out a common term of
`\(\sin^2 x\):`

`\[(8\sin^2 x(2 + \sin^2 x) + \sin^4 x)/(\ln x).\]`

Now, factor
`\(\sin^2 x\)` from the first term in the numerator:

`\[(\sin^2 x(8(2 + \sin^2 x) + \sin^2 x))/(\ln x).\]`

Distribute
`\(\sin^2 x\)` in the brackets:

`\[(8\sin^2 x + 16\sin^2 x + \sin^4 x)/(\ln x).\]`

Combine like terms:

`\[(24\sin^2 x + \sin^4 x)/(\ln x).\]`

This expression represents the simplified form of
`\(((4\cos x + \sin^2 x)^2)/(\ln x)\)` in terms of trigonometric functions and the natural logarithm.