If $\alpha + \beta = \frac{\pi}{2}$ and $\beta + \gamma = \alpha$, the…

Question

If $\alpha + \beta = \frac{\pi}{2}$ and $\beta + \gamma = \alpha$, then $	an \alpha$ equals

Accepted Answer

C. $ an \beta + 2 an \gamma$ Given $\alpha+\beta=\frac{\pi}{2}$, we get $\alpha=\frac\pi2-\beta$.

Also, $\beta+\gamma=\alpha$. Substituting for $\alpha$ gives $$\beta+\gamma=\frac\pi2-\beta$$ so $$\gamma=\frac\pi2-2\beta.$$

Now $$ an\alpha= an(\beta+\gamma).$$ Using the addition formula, $$ an\alpha=\frac{ an\beta+ an\gamma}{1- an\beta an\gamma}.$$

From $\alpha+\beta=\frac\pi2$, we have $$ an\alpha an\beta=1.$$ Let $x= an\beta$ and $y= an\gamma$. Then $$\frac1x=\frac{x+y}{1-xy}.$$ This gives $$1-xy=x^2+xy$$ and hence $$1=x^2+2xy.$$ Dividing by $x$ gives $$\frac1x=x+2y.$$ Therefore $$ an\alpha= an\beta+2 an\gamma.$$

This matches option $ ext{C}$.

Answer

2 $(	an \beta + 	an \gamma)$

Answer

$	an \beta + 	an \gamma$

Answer

$2 	an \beta + 	an \gamma$

If $\alpha + \beta = \frac{\pi}{2}$ and $\beta + \gamma = \alpha$, then $\tan \alpha$ equals