If `(sin theta+ cosec theta)^2+(cos theta+sec theta)^2=k+tan^2 theta+cot^2 theta`, then the value of k is equal to:

Options

A. 5
B. 7
C. 2
D. 9

Answer

To solve the equation \((\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = k + \tan^2 \theta + \cot^2 \theta\), we will break it down step by step.

### Step 1: Expand the left-hand side
We start by expanding the squares on 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:
\[ 2\sin \theta \csc \theta = 2 \quad \text{(because \(\sin \theta \cdot \csc \theta = 1\))} \]
And \(\csc^2 \theta = 1 + \cot^2 \theta\). Thus,
\[ (\sin \theta + \csc \theta)^2 = \sin^2 \theta + 2 + (1 + \cot^2 \theta) = \sin^2 \theta + 3 + \cot^2 \theta \]

Now, for \((\cos \theta + \sec \theta)^2\):
\[ (\cos \theta + \sec \theta)^2 = \cos^2 \theta + 2\cos \theta \sec \theta + \sec^2 \theta \]
Similarly, since \(\sec \theta = \frac{1}{\cos \theta}\):
\[ 2\cos \theta \sec \theta = 2 \]
And \(\sec^2 \theta = 1 + \tan^2 \theta\). Thus,
\[ (\cos \theta + \sec \theta)^2 = \cos^2 \theta + 2 + (1 + \tan^2 \theta) = \cos^2 \theta + 3 + \tan^2 \theta \]

### Step 2: Combine the results
Now, we combine both expansions:
\[ (\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = (\sin^2 \theta + 3 + \cot^2 \theta) + (\cos^2 \theta + 3 + \tan^2 \theta) \]
Combining like terms:
\[ = (\sin^2 \theta + \cos^2 \theta) + 6 + (\tan^2 \theta + \cot^2 \theta) \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ = 1 + 6 + (\tan^2 \theta + \cot^2 \theta) = 7 + \tan^2 \theta + \cot^2 \theta \]

### Step 3: Set the equation equal to the right-hand side
Now we have:
\[ 7 + \tan^2 \theta + \cot^2 \theta = k + \tan^2 \theta + \cot^2 \theta \]

### Step 4: Solve for \(k\)
To find \(k\), we can subtract \(\tan^2 \theta + \cot^2 \theta\) from both sides:
\[ 7 = k \]

Thus, the value of \(k\) is:
\[ \boxed{7} \]