`cos^(2)10^(@)+cos^(2)50^(@)+cos^(2)70^(@)=?`

Options

A. a) `(1)/(2)`
B. b) 1
C. c) `(3)/(2)`
D. d) 2

Answer

To solve the problem \( \cos^2 10^\circ + \cos^2 50^\circ + \cos^2 70^\circ \), we can follow these steps:

### Step 1: Rewrite the angles
We start by rewriting \( \cos^2 50^\circ \) and \( \cos^2 70^\circ \) in terms of \( \cos^2 10^\circ \):
- \( \cos^2 50^\circ = \cos^2(60^\circ - 10^\circ) \)
- \( \cos^2 70^\circ = \cos^2(60^\circ + 10^\circ) \)

### Step 2: Apply the cosine square identity
Using the cosine of sum and difference identities:
- \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)

We can express:
- \( \cos^2(60^\circ - 10^\circ) = \left( \cos 60^\circ \cos 10^\circ + \sin 60^\circ \sin 10^\circ \right)^2 \)
- \( \cos^2(60^\circ + 10^\circ) = \left( \cos 60^\circ \cos 10^\circ - \sin 60^\circ \sin 10^\circ \right)^2 \)

### Step 3: Substitute known values
We know:
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)

Substituting these values:
1. For \( \cos^2 50^\circ \):
\[ \cos^2(60^\circ - 10^\circ) = \left( \frac{1}{2} \cos 10^\circ + \frac{\sqrt{3}}{2} \sin 10^\circ \right)^2 \]
2. For \( \cos^2 70^\circ \):
\[ \cos^2(60^\circ + 10^\circ) = \left( \frac{1}{2} \cos 10^\circ - \frac{\sqrt{3}}{2} \sin 10^\circ \right)^2 \]

### Step 4: Expand the squares
Now we expand both squares:
1. \( \cos^2(60^\circ - 10^\circ) \):
\[ = \frac{1}{4} \cos^2 10^\circ + \frac{\sqrt{3}}{2} \cos 10^\circ \sin 10^\circ + \frac{3}{4} \sin^2 10^\circ \]
2. \( \cos^2(60^\circ + 10^\circ) \):
\[ = \frac{1}{4} \cos^2 10^\circ - \frac{\sqrt{3}}{2} \cos 10^\circ \sin 10^\circ + \frac{3}{4} \sin^2 10^\circ \]

### Step 5: Combine all terms
Now we combine \( \cos^2 10^\circ + \cos^2 50^\circ + \cos^2 70^\circ \):
\[ \cos^2 10^\circ + \left( \frac{1}{4} \cos^2 10^\circ + \frac{\sqrt{3}}{2} \cos 10^\circ \sin 10^\circ + \frac{3}{4} \sin^2 10^\circ \right) + \left( \frac{1}{4} \cos^2 10^\circ - \frac{\sqrt{3}}{2} \cos 10^\circ \sin 10^\circ + \frac{3}{4} \sin^2 10^\circ \right) \]

### Step 6: Simplify the expression
Combining like terms:
- The \( \cos^2 10^\circ \) terms give \( \cos^2 10^\circ + \frac{1}{4} \cos^2 10^\circ + \frac{1}{4} \cos^2 10^\circ = \frac{3}{2} \cos^2 10^\circ \)
- The \( \sin^2 10^\circ \) terms give \( \frac{3}{4} \sin^2 10^\circ + \frac{3}{4} \sin^2 10^\circ = \frac{3}{2} \sin^2 10^\circ \)
- The mixed terms cancel out.

Thus, we have:
\[ \frac{3}{2} (\cos^2 10^\circ + \sin^2 10^\circ) = \frac{3}{2} \cdot 1 = \frac{3}{2} \]

### Final Answer
The final answer is:
\[ \cos^2 10^\circ + \cos^2 50^\circ + \cos^2 70^\circ = \frac{3}{2} \]