What is the simplified value of `(3)/(cosec^(2)theta)+(5)/(1+tan^(2)theta)-2cos^(2)theta`?

Options

A. 3
B. 4
C. 5
D. 7

Answer

To simplify the expression \(\frac{3}{\csc^2 \theta} + \frac{5}{1 + \tan^2 \theta} - 2 \cos^2 \theta\), we will follow these steps:

### Step 1: Rewrite the terms using trigonometric identities
We know that:
- \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\)
- \(1 + \tan^2 \theta = \sec^2 \theta\)
- \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\)

Thus, we can rewrite the expression as:
\[ \frac{3}{\csc^2 \theta} = 3 \sin^2 \theta \]
\[ \frac{5}{1 + \tan^2 \theta} = 5 \cos^2 \theta \]
So, the expression becomes:
\[ 3 \sin^2 \theta + 5 \cos^2 \theta - 2 \cos^2 \theta \]

### Step 2: Combine like terms
Now, we can combine the terms involving \(\cos^2 \theta\):
\[ 3 \sin^2 \theta + (5 \cos^2 \theta - 2 \cos^2 \theta) = 3 \sin^2 \theta + 3 \cos^2 \theta \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can rewrite \(3 \sin^2 \theta + 3 \cos^2 \theta\) as:
\[ 3(\sin^2 \theta + \cos^2 \theta) = 3 \cdot 1 = 3 \]

### Final Answer
The simplified value of the expression is:
\[ \boxed{3} \]