If `(sintheta+cosectheta)^(2)+(costheta+sectheta)^(2)=K+tan^(2)theta+cot^(2)theta`, then K = ?

Options

A. 6
B. 3
C. 0
D. 7

Answer

To solve the equation \((\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = K + \tan^2 \theta + \cot^2 \theta\), we will simplify the left-hand side (LHS) step by step and find the value of \(K\).

### Step-by-Step Solution:

1. **Expand the LHS**:
\[ (\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 \]
Expanding each square:
\[ = (\sin^2 \theta + 2\sin \theta \csc \theta + \csc^2 \theta) + (\cos^2 \theta + 2\cos \theta \sec \theta + \sec^2 \theta) \]

2. **Simplify the terms**:
We know that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\). Thus,
\[ 2\sin \theta \csc \theta = 2\sin \theta \cdot \frac{1}{\sin \theta} = 2 \]
and
\[ 2\cos \theta \sec \theta = 2\cos \theta \cdot \frac{1}{\cos \theta} = 2 \]
Therefore, the LHS becomes:
\[ = \sin^2 \theta + \csc^2 \theta + \cos^2 \theta + \sec^2 \theta + 2 + 2 \]
\[ = \sin^2 \theta + \cos^2 \theta + \csc^2 \theta + \sec^2 \theta + 4 \]

3. **Use trigonometric identities**:
We know that \(\sin^2 \theta + \cos^2 \theta = 1\).
Also, using the identities:
\[ \csc^2 \theta = 1 + \cot^2 \theta \quad \text{and} \quad \sec^2 \theta = 1 + \tan^2 \theta \]
Substituting these into the equation gives:
\[ = 1 + (1 + \cot^2 \theta) + (1 + \tan^2 \theta) + 4 \]
\[ = 1 + 1 + 1 + 4 + \tan^2 \theta + \cot^2 \theta \]
\[ = 7 + \tan^2 \theta + \cot^2 \theta \]

4. **Set the LHS equal to the RHS**:
Now we have:
\[ 7 + \tan^2 \theta + \cot^2 \theta = K + \tan^2 \theta + \cot^2 \theta \]

5. **Solve for \(K\)**:
By comparing both sides, we can see that:
\[ K = 7 \]

Thus, the value of \(K\) is \(\boxed{7}\).