The value of `2 cos10^(@)+sin 100^(@)+sin 1000^(@)+sin 10000^(@)` is

Options

A. 0
B. `sin 10^(@)`
C. `cos 10^(@)`
D. `-1`

Explanation

To find the value of \(2 \cos 10^\circ + \sin 100^\circ + \sin 1000^\circ + \sin 10000^\circ\), we can simplify each term step by step.

### Step 1: Rewrite the sine terms
We start by rewriting the sine terms using known identities:

1. \(\sin 100^\circ\) can be rewritten as:
\[ \sin 100^\circ = \sin(90^\circ + 10^\circ) = \cos 10^\circ \]

2. For \(\sin 1000^\circ\):
\[ 1000^\circ = 2 \times 360^\circ + 280^\circ \Rightarrow \sin 1000^\circ = \sin 280^\circ = \sin(360^\circ - 80^\circ) = -\sin 80^\circ \]

3. For \(\sin 10000^\circ\):
\[ 10000^\circ = 27 \times 360^\circ + 280^\circ \Rightarrow \sin 10000^\circ = \sin 280^\circ = -\sin 80^\circ \]

### Step 2: Substitute back into the expression
Now we substitute these back into the original expression:
\[ 2 \cos 10^\circ + \sin 100^\circ + \sin 1000^\circ + \sin 10000^\circ = 2 \cos 10^\circ + \cos 10^\circ - \sin 80^\circ - \sin 80^\circ \]

### Step 3: Combine like terms
Now we can combine the terms:
\[ = 2 \cos 10^\circ + \cos 10^\circ - 2 \sin 80^\circ \]
\[ = 3 \cos 10^\circ - 2 \sin 80^\circ \]

### Step 4: Simplify \(\sin 80^\circ\)
Using the identity \(\sin 80^\circ = \cos 10^\circ\), we can substitute:
\[ = 3 \cos 10^\circ - 2 \cos 10^\circ \]
\[ = (3 - 2) \cos 10^\circ = 1 \cos 10^\circ \]

### Final Answer
Thus, the final value is:
\[ \cos 10^\circ \]