IIT-JEE 2001 ScreeningQuestion 1
If $\alpha+\beta=\pi/2$ and $\beta+\gamma=\alpha$, then $\tan\alpha$ equals
Trigonometric Ratio & Identities
Solution
Concept. $\alpha+\beta=\frac\pi2$ gives $\tan\alpha\tan\beta=1$; then $\tan\gamma=\tan(\alpha-\beta)$ simplifies. Formulas used. (1) $\tan(\alpha-\beta)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$Step 1 - Product. $\beta=\frac\pi2-\alpha$, so $\tan\beta=\cot\alpha$ and $\tan\alpha\tan\beta=1$. Step 2 - Use (1). $\gamma=\alpha-\beta$, so $$\tan\gamma=\frac{\tan\alpha-\tan\beta}{2}.$$ Step 3 - Rearrange. $\tan\alpha=\tan\beta+2\tan\gamma$. Answer: (C)