If `cotA + "cosec"A=3` and A is an acute angle then the value of cosA is:

Options

A. 1
B. `1/2`
C. `3/4`
D. `4/5`

Answer

To solve the problem, we start with the given equation:

1. **Given Equation**:
\[ \cot A + \csc A = 3 \]

2. **Using Trigonometric Identity**:
We know the identity:
\[ \cot^2 A + 1 = \csc^2 A \]
Rearranging this gives:
\[ \cot^2 A = \csc^2 A - 1 \]

3. **Substituting**:
From the given equation, we can express \(\csc A\) in terms of \(\cot A\):
\[ \csc A = 3 - \cot A \]
Now, substituting this into the identity:
\[ \cot^2 A = (3 - \cot A)^2 - 1 \]

4. **Expanding the Equation**:
Expanding the right-hand side:
\[ \cot^2 A = (9 - 6\cot A + \cot^2 A) - 1 \]
Simplifying gives:
\[ \cot^2 A = 8 - 6\cot A + \cot^2 A \]

5. **Cancelling \(\cot^2 A\)**:
Subtract \(\cot^2 A\) from both sides:
\[ 0 = 8 - 6\cot A \]
Rearranging gives:
\[ 6\cot A = 8 \]
Thus:
\[ \cot A = \frac{8}{6} = \frac{4}{3} \]

6. **Finding Sides of the Triangle**:
In a right triangle, if \(\cot A = \frac{\text{adjacent}}{\text{opposite}} = \frac{4}{3}\), we can label:
- Adjacent side = 4
- Opposite side = 3

7. **Finding Hypotenuse**:
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
\[ \text{Hypotenuse}^2 = 4^2 + 3^2 = 16 + 9 = 25 \]
Thus:
\[ \text{Hypotenuse} = \sqrt{25} = 5 \]

8. **Finding \(\cos A\)**:
We know:
\[ \cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5} \]

9. **Final Answer**:
Therefore, the value of \(\cos A\) is:
\[ \cos A = \frac{4}{5} \]