Prove that `3"cos"^2 (pi)/(4)+sec^(2)(pi/3)+5 "tan"^(2)(pi)/(3)=(41)/(2)`.

Answer

To prove that

\[ 3 \cos^2 \left( \frac{\pi}{4} \right) + \sec^2 \left( \frac{\pi}{3} \right) + 5 \tan^2 \left( \frac{\pi}{3} \right) = \frac{41}{2} \]

we will calculate each term step by step.

### Step 1: Calculate \( \cos^2 \left( \frac{\pi}{4} \right) \)

We know that:

\[ \cos \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} \]

Thus,

\[ \cos^2 \left( \frac{\pi}{4} \right) = \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2} \]

### Step 2: Calculate \( 3 \cos^2 \left( \frac{\pi}{4} \right) \)

Now, substituting the value we found:

\[ 3 \cos^2 \left( \frac{\pi}{4} \right) = 3 \times \frac{1}{2} = \frac{3}{2} \]

### Step 3: Calculate \( \sec^2 \left( \frac{\pi}{3} \right) \)

We know that:

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

Thus,

\[ \sec^2 \left( \frac{\pi}{3} \right) = 2^2 = 4 \]

### Step 4: Calculate \( \tan^2 \left( \frac{\pi}{3} \right) \)

We know that:

\[ \tan \left( \frac{\pi}{3} \right) = \sqrt{3} \]

Thus,

\[ \tan^2 \left( \frac{\pi}{3} \right) = \left( \sqrt{3} \right)^2 = 3 \]

### Step 5: Calculate \( 5 \tan^2 \left( \frac{\pi}{3} \right) \)

Now substituting the value we found:

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

### Step 6: Combine all the values

Now we can combine all the calculated values:

\[ 3 \cos^2 \left( \frac{\pi}{4} \right) + \sec^2 \left( \frac{\pi}{3} \right) + 5 \tan^2 \left( \frac{\pi}{3} \right) = \frac{3}{2} + 4 + 15 \]

### Step 7: Convert all terms to a common denominator

To add these fractions, we convert all terms to have a common denominator of 2:

\[ \frac{3}{2} + 4 + 15 = \frac{3}{2} + \frac{8}{2} + \frac{30}{2} = \frac{3 + 8 + 30}{2} = \frac{41}{2} \]

### Conclusion

Thus, we have shown that:

\[ 3 \cos^2 \left( \frac{\pi}{4} \right) + \sec^2 \left( \frac{\pi}{3} \right) + 5 \tan^2 \left( \frac{\pi}{3} \right) = \frac{41}{2} \]

Hence proved.

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