Evaluate : `sin50^(@)cos40^(@)+cos40^(@)+cos50^(@)sin40^(@)`

Answer

To evaluate the expression \( \sin 50^\circ \cos 40^\circ + \cos 40^\circ + \cos 50^\circ \sin 40^\circ \), we can follow these steps:

### Step 1: Identify the angles
Let \( A = 50^\circ \) and \( B = 40^\circ \).

### Step 2: Rewrite the expression
The expression can be rewritten as:
\[ \sin A \cos B + \cos B + \cos A \sin B \]

### Step 3: Group terms
We can group the first and the last terms:
\[ (\sin A \cos B + \cos A \sin B) + \cos B \]

### Step 4: Apply the sine addition formula
Recall the sine addition formula:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
Thus, we can rewrite the expression as:
\[ \sin(A + B) + \cos B \]

### Step 5: Substitute the values of A and B
Now substituting \( A + B = 50^\circ + 40^\circ = 90^\circ \):
\[ \sin 90^\circ + \cos 40^\circ \]

### Step 6: Evaluate \( \sin 90^\circ \)
We know that:
\[ \sin 90^\circ = 1 \]
So the expression simplifies to:
\[ 1 + \cos 40^\circ \]

### Step 7: Final expression
Thus, the final expression is:
\[ 1 + \cos 40^\circ \]

### Step 8: Further simplification
Using the identity \( 1 + \cos \theta = 2 \cos^2(\theta/2) \):
\[ 1 + \cos 40^\circ = 2 \cos^2(20^\circ) \]

### Final Answer
The final answer is:
\[ 2 \cos^2(20^\circ) \]

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