If : `(sintheta+csctheta)^(2)+(costheta+sectheta)^(2)=tan^(2)theta+cot^(2)theta+k,`
then : k =

Options

A. 1
B. 3
C. 5
D. 7

Answer

To solve the equation

\[ (\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = \tan^2 \theta + \cot^2 \theta + k, \]

we will simplify both sides step by step to find the value of \( k \).

### Step 1: Expand the left-hand side

We start by expanding the left-hand side:

\[ (\sin \theta + \csc \theta)^2 = \sin^2 \theta + 2 \sin \theta \csc \theta + \csc^2 \theta \]

Since \(\csc \theta = \frac{1}{\sin \theta}\), we have:

\[ \sin^2 \theta + 2 + \csc^2 \theta \]

Similarly, for \((\cos \theta + \sec \theta)^2\):

\[ (\cos \theta + \sec \theta)^2 = \cos^2 \theta + 2 \cos \theta \sec \theta + \sec^2 \theta \]

Since \(\sec \theta = \frac{1}{\cos \theta}\), we have:

\[ \cos^2 \theta + 2 + \sec^2 \theta \]

### Step 2: Combine the results

Now, we combine both expansions:

\[ (\sin^2 \theta + \csc^2 \theta + 2) + (\cos^2 \theta + \sec^2 \theta + 2) \]

This simplifies to:

\[ \sin^2 \theta + \cos^2 \theta + \csc^2 \theta + \sec^2 \theta + 4 \]

Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can substitute:

\[ 1 + \csc^2 \theta + \sec^2 \theta + 4 \]

### Step 3: Substitute identities for \(\csc^2 \theta\) and \(\sec^2 \theta\)

We know that:

\[ \csc^2 \theta = 1 + \cot^2 \theta \quad \text{and} \quad \sec^2 \theta = 1 + \tan^2 \theta \]

Substituting these into the equation gives us:

\[ 1 + (1 + \cot^2 \theta) + (1 + \tan^2 \theta) + 4 \]

This simplifies to:

\[ 1 + 1 + 1 + 4 + \tan^2 \theta + \cot^2 \theta = 7 + \tan^2 \theta + \cot^2 \theta \]

### Step 4: Set the equation equal to the right-hand side

Now we have:

\[ 7 + \tan^2 \theta + \cot^2 \theta = \tan^2 \theta + \cot^2 \theta + k \]

### Step 5: Solve for \( k \)

By comparing both sides, we can see that:

\[ k = 7 \]

Thus, the value of \( k \) is

\[ \boxed{7} \]