Question

Find the value ${\left(tan\right(22\frac{1}{2})}^{\circ }\times (\sqrt{2}+1)$

Answer 1

We want to find tan (22 $\frac{1}{2}$). we know (22 $\frac{1}{2}$) = $\frac{45}{2}$. it is easy to guess that we will be using

trigonometric ratios of ${45}^{\circ }$ to find tan 22$\frac{1}{2}$ now we have to think of some formula which connects

tan 22$\frac{1}{2}$ and tan 45.
We know tan2A = $\frac{2tanA}{1-ta{n}^{2}A}$
If A = 22$\frac{1}{2}$ in this equation, we will get a quadratic in tan 22$\frac{1}{2}$
$\Rightarrow$ tan45 = 1 = $\frac{2tan22\frac{1}{2}}{1-ta{n}^{2}22\frac{1}{2}}$
$\Rightarrow$ $ta{n}^{2}22\frac{1}{2}+2tan22\frac{1}{2}-1=0$

$\Rightarrow$ tan 22 $\frac{1}{2}=\frac{-2\pm \sqrt{4+4}}{2}$

= $\frac{-2\pm \sqrt{4+4}}{2}$

= $\frac{-2\pm 2\sqrt{2}}{2}$

= -1 $\pm$± $\sqrt{2}$

$Tan22\frac{1}{2}$ is +ve

$\Rightarrow$ $tan22\frac{1}{2}=-1+\sqrt{2}=\sqrt{2}-1$

$\Rightarrow$ $\sqrt{2}+1)tan22\frac{1}{2}=(\sqrt{2}+1\left)\right(\sqrt{2}-1)=1$

Key steps (1) 22 $\frac{1}{2}=\frac{45}{2}$

(2) Guessing the relation which has tan 22 $\frac{1}{2}$ and tan 45 .