In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.
Step-by-Step Solution
Key Concept: Use the definition of tangent in a right triangle (tan A = opposite/adjacent = sin A / cos A) and the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\). From tan A = 1 we get \(\sin A = \cos A\); substituting into the identity gives the exact values of \(\sin A\) and \(\cos A\), which can then be used to evaluate \(2\sin A\cos A\).
1. In \(\triangle ABC\) right‑angled at \(B\), the trigonometric ratios are defined as:
$$\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin A}{\cos A}.$$
2. Given \(\tan A = 1\), we have
$$\frac{\sin A}{\cos A} = 1 \quad\Rightarrow\quad \sin A = \cos A. $$
3. Use the fundamental identity for any angle:
$$\sin^2 A + \cos^2 A = 1.$$
4. Substitute \(\sin A = \cos A\) into the identity:
$$\sin^2 A + \sin^2 A = 1 \;\Rightarrow\; 2\sin^2 A = 1 \;\Rightarrow\; \sin^2 A = \frac{1}{2}.$$
5. Hence
$$\sin A = \cos A = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.$$
6. Now evaluate the expression to be verified:
$$2\sin A \cos A = 2\left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)
= 2\times \frac{2}{4}
= 1.$$
7. Therefore, the required identity is verified:
$$2\sin A \cos A = 1.$$
Correct Answer: Verified: 2 sin A cos A = 1