If `cos A=4/5`, then the value of `tanA` is

Options

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

Explanation

To find the value of \( \tan A \) given that \( \cos A = \frac{4}{5} \), we can follow these steps:

### Step 1: Recall the relationship between sine, cosine, and tangent.
We know that:
\[ \tan A = \frac{\sin A}{\cos A} \]

### Step 2: Find \( \sin A \) using the Pythagorean identity.
The Pythagorean identity states:
\[ \sin^2 A + \cos^2 A = 1 \]
We can rearrange this to find \( \sin A \):
\[ \sin^2 A = 1 - \cos^2 A \]

### Step 3: Substitute the value of \( \cos A \).
Given \( \cos A = \frac{4}{5} \), we can substitute this into the equation:
\[ \sin^2 A = 1 - \left(\frac{4}{5}\right)^2 \]
Calculating \( \left(\frac{4}{5}\right)^2 \):
\[ \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]
Now substitute this back:
\[ \sin^2 A = 1 - \frac{16}{25} \]
Convert 1 to a fraction with a denominator of 25:
\[ 1 = \frac{25}{25} \]
So,
\[ \sin^2 A = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]

### Step 4: Take the square root to find \( \sin A \).
\[ \sin A = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 5: Substitute \( \sin A \) and \( \cos A \) into the tangent formula.
Now we can find \( \tan A \):
\[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{3}{5}}{\frac{4}{5}} \]

### Step 6: Simplify the expression.
When we divide two fractions, we multiply by the reciprocal:
\[ \tan A = \frac{3}{5} \times \frac{5}{4} = \frac{3}{4} \]

### Final Answer:
Thus, the value of \( \tan A \) is \( \frac{3}{4} \).

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