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