Find the value of `(tan 40^@ sec 50^@)/(cot 50^@ cosec 40^@) + sin 50^@ cos 40^@ + cos^2 50^@ + tan 30^@`

Options

A. `sqrt""3`
B. `sqrt""3 + 2`
C. `(2 + sqrt(2))/(sqrt(3))`
D. `(2sqrt(3) + 1)/(sqrt(3))`

Answer

To solve the expression \((\tan 40^\circ \sec 50^\circ) / (\cot 50^\circ \csc 40^\circ) + \sin 50^\circ \cos 40^\circ + \cos^2 50^\circ + \tan 30^\circ\), we will break it down step-by-step.

### Step 1: Rewrite Trigonometric Functions
Using the identities:
- \(\tan(90^\circ - \theta) = \cot \theta\)
- \(\sec(90^\circ - \theta) = \csc \theta\)
- \(\cos(90^\circ - \theta) = \sin \theta\)

We can rewrite \(\tan 40^\circ\) and \(\sec 50^\circ\):
\[ \tan 40^\circ = \cot 50^\circ \quad \text{and} \quad \sec 50^\circ = \csc 40^\circ \]

### Step 2: Substitute in the Expression
Now substituting these identities into the expression:
\[ \frac{\tan 40^\circ \sec 50^\circ}{\cot 50^\circ \csc 40^\circ} = \frac{\cot 50^\circ \csc 40^\circ}{\cot 50^\circ \csc 40^\circ} = 1 \]

### Step 3: Simplify the Remaining Terms
Now we simplify the remaining terms:
\[ 1 + \sin 50^\circ \cos 40^\circ + \cos^2 50^\circ + \tan 30^\circ \]
We know that:
\(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

### Step 4: Simplify \(\sin 50^\circ \cos 40^\circ\)
Using the identity \(\sin 50^\circ = \cos 40^\circ\):
\[ \sin 50^\circ \cos 40^\circ = \cos 40^\circ \cos 40^\circ = \cos^2 40^\circ \]

### Step 5: Combine Terms
Now we can combine:
\[ 1 + \cos^2 40^\circ + \cos^2 50^\circ + \frac{1}{\sqrt{3}} \]

Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), we know:
\(\cos^2 40^\circ + \cos^2 50^\circ = 1 - \sin^2 40^\circ + 1 - \sin^2 50^\circ\)

### Step 6: Final Calculation
Since \(\sin^2 40^\circ + \sin^2 50^\circ = 1\):
\[ 1 + 1 + \frac{1}{\sqrt{3}} = 2 + \frac{1}{\sqrt{3}} \]

### Step 7: Find a Common Denominator
To express \(2\) in terms of \(\sqrt{3}\):
\[ 2 = \frac{2\sqrt{3}}{\sqrt{3}} \]
Thus:
\[ 2 + \frac{1}{\sqrt{3}} = \frac{2\sqrt{3} + 1}{\sqrt{3}} \]

### Final Answer
The final value of the expression is:
\[ \frac{2\sqrt{3} + 1}{\sqrt{3}} \]