If ` 4 cos ^(2) A -3 =0 and 0^(@) le A le 90 ^(@) ` find :
` tan ^(2) A+ cos ^(2) A `

Answer

To solve the problem step by step, we will follow the given equation and find the required values.

### Step 1: Solve the equation for cos²A
Given the equation:
\[ 4 \cos^2 A - 3 = 0 \]

Rearranging this gives:
\[ 4 \cos^2 A = 3 \]

Dividing both sides by 4:
\[ \cos^2 A = \frac{3}{4} \]

### Step 2: Find the value of cos A
Taking the square root of both sides:
\[ \cos A = \sqrt{\frac{3}{4}} \]
\[ \cos A = \frac{\sqrt{3}}{2} \]

### Step 3: Determine the angle A
From trigonometric values, we know:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus, we have:
\[ A = 30^\circ \]

### Step 4: Find tan²A
Using the angle A, we find:
\[ \tan A = \tan 30^\circ = \frac{1}{\sqrt{3}} \]

Now, squaring this gives:
\[ \tan^2 A = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 5: Find cos²A
From Step 1, we already found:
\[ \cos^2 A = \frac{3}{4} \]

### Step 6: Calculate tan²A + cos²A
Now we add the two results:
\[ \tan^2 A + \cos^2 A = \frac{1}{3} + \frac{3}{4} \]

To add these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12.

Converting each fraction:
\[ \frac{1}{3} = \frac{4}{12} \]
\[ \frac{3}{4} = \frac{9}{12} \]

Now, adding them together:
\[ \tan^2 A + \cos^2 A = \frac{4}{12} + \frac{9}{12} = \frac{13}{12} \]

### Final Answer
Thus, the final result is:
\[ \tan^2 A + \cos^2 A = \frac{13}{12} \]

---