Evaluate (ii) `cos{tan^(-1) (3/4)}`

Answer

To evaluate \( \cos(\tan^{-1}(3/4)) \), we can follow these steps:

### Step 1: Let \( \theta = \tan^{-1}(3/4) \)
This means that \( \tan(\theta) = \frac{3}{4} \).

**Hint:** Remember that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

### Step 2: Form a right triangle
In a right triangle, if we let the opposite side (perpendicular) be 3 and the adjacent side (base) be 4, we can find the hypotenuse using the Pythagorean theorem.

**Hint:** The Pythagorean theorem states that \( h^2 = a^2 + b^2 \), where \( h \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

### Step 3: Calculate the hypotenuse
Using the sides we have:
\[ h = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

**Hint:** Make sure to square each side correctly and then add them before taking the square root.

### Step 4: Find \( \cos(\theta) \)
Now that we have the lengths of all sides of the triangle, we can find \( \cos(\theta) \):
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5} \]

**Hint:** The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

### Step 5: Conclusion
Thus, we find that:
\[ \cos(\tan^{-1}(3/4)) = \frac{4}{5} \]

**Final Answer:** \( \frac{4}{5} \)