What is the value of expression `cos^(2)((pi)/(8))+4cos^(2)((pi)/(4))-sec((pi)/(3))+5tan^(2)((pi)/(3))+sin^(2)((pi)/(8))` ?

Options

A. `8`
B. `10`
C. `16`
D. `18`

Answer

To solve the expression \( \cos^2\left(\frac{\pi}{8}\right) + 4\cos^2\left(\frac{\pi}{4}\right) - \sec\left(\frac{\pi}{3}\right) + 5\tan^2\left(\frac{\pi}{3}\right) + \sin^2\left(\frac{\pi}{8}\right) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \cos^2\left(\frac{\pi}{4}\right) \)
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \implies \cos^2\left(\frac{\pi}{4}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 2: Evaluate \( 4\cos^2\left(\frac{\pi}{4}\right) \)
\[ 4\cos^2\left(\frac{\pi}{4}\right) = 4 \times \frac{1}{2} = 2 \]

### Step 3: Evaluate \( \sec\left(\frac{\pi}{3}\right) \)
\[ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 4: Evaluate \( \tan^2\left(\frac{\pi}{3}\right) \)
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \implies \tan^2\left(\frac{\pi}{3}\right) = (\sqrt{3})^2 = 3 \]

### Step 5: Evaluate \( 5\tan^2\left(\frac{\pi}{3}\right) \)
\[ 5\tan^2\left(\frac{\pi}{3}\right) = 5 \times 3 = 15 \]

### Step 6: Evaluate \( \sin^2\left(\frac{\pi}{8}\right) \)
Using the identity \( \sin^2\theta + \cos^2\theta = 1 \):
\[ \sin^2\left(\frac{\pi}{8}\right) = 1 - \cos^2\left(\frac{\pi}{8}\right) \]
However, we will keep \( \sin^2\left(\frac{\pi}{8}\right) \) as is for now.

### Step 7: Combine all the evaluated parts
Now, we can combine all the parts:
\[ \cos^2\left(\frac{\pi}{8}\right) + \sin^2\left(\frac{\pi}{8}\right) + 4\cos^2\left(\frac{\pi}{4}\right) - \sec\left(\frac{\pi}{3}\right) + 5\tan^2\left(\frac{\pi}{3}\right) \]
This simplifies to:
\[ 1 + 2 - 2 + 15 \]

### Step 8: Final Calculation
\[ 1 + 2 - 2 + 15 = 16 \]

Thus, the value of the expression is \( \boxed{16} \).

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