If `2sintheta+costheta=(7)/(3)` then the value of `(tan^(2)theta-sec^(2)theta)` is

Options

A. 0
B. `-1`
C. `(3)/(7)`
D. `(7)/(3)`

Answer

To solve the problem, we start with the given equation:

\[ 2 \sin \theta + \cos \theta = \frac{7}{3} \]

We need to find the value of:

\[ \tan^2 \theta - \sec^2 \theta \]

### Step 1: Use the identity for \(\tan^2 \theta\) and \(\sec^2 \theta\)

We know the trigonometric identity:

\[ \tan^2 \theta + 1 = \sec^2 \theta \]

From this, we can express \(\tan^2 \theta\) in terms of \(\sec^2 \theta\):

\[ \tan^2 \theta = \sec^2 \theta - 1 \]

### Step 2: Substitute \(\tan^2 \theta\) into the expression

Now, substituting \(\tan^2 \theta\) into our expression:

\[ \tan^2 \theta - \sec^2 \theta = (\sec^2 \theta - 1) - \sec^2 \theta \]

### Step 3: Simplify the expression

Now, simplifying the expression:

\[ \tan^2 \theta - \sec^2 \theta = \sec^2 \theta - 1 - \sec^2 \theta \]

The \(\sec^2 \theta\) terms cancel out:

\[ \tan^2 \theta - \sec^2 \theta = -1 \]

### Final Answer

Thus, the value of \(\tan^2 \theta - \sec^2 \theta\) is:

\[ \boxed{-1} \]

---