If \( \tan A = \frac{3}{4}\), then \(\sin A \cos A= \frac{12}{25}\)

To prove \(\sin A \cos A= \frac{12}{25}\), if \( \tan A = \frac{3}{4}\)

Solution:

Step 1: Find opposite and adjacent side

Given \( \tan A = \frac{3}{4}= \frac{3k}{4k}\), where \(k\) is any non-zero integer.

As we know \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}\).

Therefore, \(\text{opposite side}= 3k, \quad \text{adjacent side} = 4k\)

Step 2: Find Hypotenuse

By Pythagoras theorem, \( \text{Hypotenuse}^2=\text{opposite side}^2+\text{adjacent side}^2\)

Therefore, \( \text{Hypotenuse}^2=(3k)^2+(4k)^2\)

\(\Rightarrow  \text{Hypotenuse}^2=9k^2+16k^2\)

\(\Rightarrow  \text{Hypotenuse}^2=25k^2\)

\(\Rightarrow  \text{Hypotenuse}=5k\)

Step 3: Find the Values of \(\sin A\) and \(\cos A\)

\(\sin A= \frac{\text{opposite side}}{\text{Hypotenuse}}\) and \(\cos A=\frac{\text{adjacent side}}{\text{Hypotenuse}}\)

\(\Rightarrow\) \(\sin A= \frac{3k}{5k}\) and \(\cos A=\frac{4k}{5k}\)

\(\Rightarrow\) \(\sin A= \frac{3}{5}\) and \(\cos A=\frac{4}{5}\)

Step 4: Find the Product of \(\sin A\) and \(\cos A\)

\(\sin A \cos A= \frac{3}{5} \times \frac{4}{5}\)

\(\Rightarrow \sin A \cos A= \frac{3\times 4}{5\times 5}\)

\(\Rightarrow \sin A \cos A= \frac{12}{25}\)

Therefore, if \(\tan A= \frac{3}{4}\), then \(\sin A \cos A = \frac{12}{25}\).