If `sin theta=cos theta`, then `2tan^(2)theta+sin^(2)theta-1=`_____

Options

A. `(-3)/2`
B. `3/2`
C. `2/3`
D. `(-2)/3`

Answer

To solve the equation \(2\tan^2\theta + \sin^2\theta - 1\) given that \(\sin\theta = \cos\theta\), we can follow these steps:

### Step 1: Identify the angle
Since \(\sin\theta = \cos\theta\), we know that this equality holds true when \(\theta = \frac{\pi}{4}\) or \(45^\circ\).

### Step 2: Calculate \(\tan\theta\)
Using the angle \(\theta = \frac{\pi}{4}\):
\[ \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1 \]
Thus,
\[ \tan^2\theta = 1^2 = 1 \]

### Step 3: Calculate \(\sin^2\theta\)
For \(\theta = \frac{\pi}{4}\):
\[ \sin\theta = \frac{1}{\sqrt{2}} \implies \sin^2\theta = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 4: Substitute values into the expression
Now, substitute \(\tan^2\theta\) and \(\sin^2\theta\) into the expression \(2\tan^2\theta + \sin^2\theta - 1\):
\[ 2\tan^2\theta + \sin^2\theta - 1 = 2(1) + \frac{1}{2} - 1 \]

### Step 5: Simplify the expression
Now simplify:
\[ = 2 + \frac{1}{2} - 1 \]
\[ = 2 - 1 + \frac{1}{2} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]

### Final Answer
Thus, the value of \(2\tan^2\theta + \sin^2\theta - 1\) is \(\frac{3}{2}\).

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