`sin 10^(@)+sin 20^(@) + sin 40 ^(@) + sin 50 =` A)`sin 15^(@) +sin 75^(@)`B)`cos 15^(@) + cos75^(@)`C)`sin 70^(@) + sin 80^(@)`D)`cos 70^(@) + cos 80^(@)`

Options

A. `sin 15^(@) +sin 75^(@)`
B. `cos 15^(@) + cos75^(@)`
C. `sin 70^(@) + sin 80^(@)`
D. `cos 70^(@) + cos 80^(@)`

Answer

To solve the expression \( \sin 10^\circ + \sin 20^\circ + \sin 40^\circ + \sin 50^\circ \), we can use the sine addition formula. The sine addition formula states that:

\[ \sin x + \sin y = 2 \sin \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \]

### Step 1: Group the terms
We can group the terms in pairs:
- Pair 1: \( \sin 10^\circ + \sin 50^\circ \)
- Pair 2: \( \sin 20^\circ + \sin 40^\circ \)

### Step 2: Apply the sine addition formula to Pair 1
For Pair 1:
\[ \sin 10^\circ + \sin 50^\circ = 2 \sin \left( \frac{10^\circ + 50^\circ}{2} \right) \cos \left( \frac{10^\circ - 50^\circ}{2} \right) \]
Calculating the angles:
\[ = 2 \sin \left( \frac{60^\circ}{2} \right) \cos \left( \frac{-40^\circ}{2} \right) = 2 \sin 30^\circ \cos (-20^\circ) \]
Since \( \sin 30^\circ = \frac{1}{2} \) and \( \cos (-\theta) = \cos \theta \):
\[ = 2 \cdot \frac{1}{2} \cdot \cos 20^\circ = \cos 20^\circ \]

### Step 3: Apply the sine addition formula to Pair 2
For Pair 2:
\[ \sin 20^\circ + \sin 40^\circ = 2 \sin \left( \frac{20^\circ + 40^\circ}{2} \right) \cos \left( \frac{20^\circ - 40^\circ}{2} \right) \]
Calculating the angles:
\[ = 2 \sin \left( \frac{60^\circ}{2} \right) \cos \left( \frac{-20^\circ}{2} \right) = 2 \sin 30^\circ \cos (-10^\circ) \]
Again, since \( \sin 30^\circ = \frac{1}{2} \):
\[ = 2 \cdot \frac{1}{2} \cdot \cos 10^\circ = \cos 10^\circ \]

### Step 4: Combine the results
Now we combine the results from both pairs:
\[ \sin 10^\circ + \sin 20^\circ + \sin 40^\circ + \sin 50^\circ = \cos 20^\circ + \cos 10^\circ \]

### Step 5: Use the cosine addition formula
Using the cosine addition formula:
\[ \cos x + \cos y = 2 \cos \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \]
Let \( x = 20^\circ \) and \( y = 10^\circ \):
\[ \cos 20^\circ + \cos 10^\circ = 2 \cos \left( \frac{30^\circ}{2} \right) \cos \left( \frac{10^\circ}{2} \right) = 2 \cos 15^\circ \cos 5^\circ \]

### Conclusion
Thus, the expression simplifies to:
\[ \sin 10^\circ + \sin 20^\circ + \sin 40^\circ + \sin 50^\circ = \sin 70^\circ + \sin 80^\circ \]

### Final Answer
The correct option is:
**C) \( \sin 70^\circ + \sin 80^\circ \)**

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