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: Write the Expression
We start with the given expression:
\[ \sin 50^\circ \cos 40^\circ + \cos 40^\circ + \cos 50^\circ \sin 40^\circ \]

### Step 2: Use Trigonometric Identities
We know that:
- \( \sin(90^\circ - \theta) = \cos \theta \)
- \( \cos(90^\circ - \theta) = \sin \theta \)

Using these identities, we can rewrite \( \cos 40^\circ \) and \( \sin 40^\circ \):
\[ \sin 50^\circ \cos 40^\circ + \cos 40^\circ + \cos 50^\circ \sin 40^\circ = \sin 50^\circ \cos 40^\circ + \cos 40^\circ + \cos 50^\circ \sin 40^\circ \]

### Step 3: Group Terms
Notice that we can group the terms involving \( \cos 40^\circ \):
\[ = \cos 40^\circ (\sin 50^\circ + 1) + \cos 50^\circ \sin 40^\circ \]

### Step 4: Apply the Sine and Cosine Addition Formula
We can also recognize that:
\[ \sin 50^\circ + \cos 50^\circ = \sin 50^\circ + \cos(90^\circ - 50^\circ) = \sin 50^\circ + \sin 40^\circ \]

### Step 5: Simplify Further
Now we can use the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Thus:
\[ \sin^2 50^\circ + \cos^2 50^\circ = 1 \]

### Step 6: Combine Results
Putting it all together:
\[ \sin^2 50^\circ + \cos^2 50^\circ + \cos 40^\circ = 1 + \cos 40^\circ \]

### Final Answer
Thus, the final result is:
\[ 1 + \cos 40^\circ \]