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

1 Answer

3 votes

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.

User Sanjeev Kumar
by
8.1k points
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