If `tanalpha=1/7,sinbeta=1/(sqrt(10)),` prove that`alpha+2beta=pi/4` , where `0

Explanation

To prove that \(\alpha + 2\beta = \frac{\pi}{4}\) given that \(\tan \alpha = \frac{1}{7}\) and \(\sin \beta = \frac{1}{\sqrt{10}}\), we will follow these steps:

### Step 1: Find \(\alpha\)
Given that \(\tan \alpha = \frac{1}{7}\), we can express this in terms of a right triangle.

Let the opposite side to angle \(\alpha\) be \(1\) and the adjacent side be \(7\). By the Pythagorean theorem, the hypotenuse \(h\) can be calculated as follows:

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