Questions: Simplify the following expression: 4 sin^2(x) + 4 cos^2(x) - 9 tan(x) cot(x) -5 4 - 9 tan(2x) 13 tan(x) 4 + 9 tan(2x) 13 None of the above.

Transcript text: Simplity the following expression: $4 \sin ^{2}(x)+4 \cos ^{2}(x)-9 \tan (x) \cot (x)$ $-5$ $4-9 \tan (2 x)$ $13 \tan (x)$ $4+9 \tan (2 x)$ 13 None of the above.

Solution

Solution Steps

To simplify the given expression, we can use trigonometric identities. The identity $\sin^2(x) + \cos^2(x) = 1$ can simplify the first part of the expression. The term $\tan(x) \cot(x)$ simplifies to 1. The expression can be rewritten and simplified using these identities.

Step 1: Simplify Using Trigonometric Identities

The given expression is:

\[ 4 \sin^2(x) + 4 \cos^2(x) - 9 \tan(x) \cot(x) \]

We can use the trigonometric identity $\sin^2(x) + \cos^2(x) = 1$ to simplify the first part of the expression:

\[ 4 (\sin^2(x) + \cos^2(x)) = 4 \times 1 = 4 \]

Step 2: Simplify $\tan(x) \cot(x)$

The term $\tan(x) \cot(x)$ simplifies to 1 because $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and $\cot(x) = \frac{\cos(x)}{\sin(x)}$, so:

\[ \tan(x) \cot(x) = \left(\frac{\sin(x)}{\cos(x)}\right) \left(\frac{\cos(x)}{\sin(x)}\right) = 1 \]

Thus, the expression becomes:

\[ 4 - 9 \times 1 = 4 - 9 = -5 \]