If `cot theta = 1` , find the value of : ` 5 tan^2 theta + 2 sin^2 theta - 3`

Answer

To solve the problem step by step, we need to find the value of the expression \( 5 \tan^2 \theta + 2 \sin^2 \theta - 3 \) given that \( \cot \theta = 1 \).

### Step 1: Find \( \tan^2 \theta \)
Since \( \cot \theta = 1 \), we can use the relationship between cotangent and tangent:
\[ \tan \theta = \frac{1}{\cot \theta} \]
Substituting the value of \( \cot \theta \):
\[ \tan \theta = \frac{1}{1} = 1 \]
Now, squaring both sides:
\[ \tan^2 \theta = 1^2 = 1 \]

### Step 2: Find \( \cos^2 \theta \)
We know the identity:
\[ \cot^2 \theta + 1 = \csc^2 \theta \]
Since \( \cot \theta = 1 \):
\[ \cot^2 \theta = 1^2 = 1 \]
Thus,
\[ 1 + 1 = \csc^2 \theta \implies \csc^2 \theta = 2 \]
Now, using the identity \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \):
\[ \sin^2 \theta = \frac{1}{\csc^2 \theta} = \frac{1}{2} \]

### Step 3: Substitute values into the expression
Now we substitute \( \tan^2 \theta \) and \( \sin^2 \theta \) into the expression:
\[ 5 \tan^2 \theta + 2 \sin^2 \theta - 3 \]
Substituting the values we found:
\[ = 5(1) + 2\left(\frac{1}{2}\right) - 3 \]
Calculating each term:
\[ = 5 + 1 - 3 \]

### Step 4: Simplify the expression
Now, we simplify:
\[ = 5 + 1 - 3 = 6 - 3 = 3 \]

### Final Answer
Thus, the value of the expression \( 5 \tan^2 \theta + 2 \sin^2 \theta - 3 \) is:
\[ \boxed{3} \]