Inverse Trigonometric Functions Question 9
Question: If $ {{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z=\pi , $ then $ \frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}= $
[MP PET 1991]
Options:
A) 0
B) 1
C) $ \frac{1}{xyz} $
D) $ xyz $
Show Answer
Answer:
Correct Answer: B
Solution:
$ {{\tan }^{-1}}(x)+{{\tan }^{-1}}(y)+{{\tan }^{-1}}(z)=\pi $
Therefore $ {{\tan }^{-1}}x+{{\tan }^{-1}}y=\pi -{{\tan }^{-1}}z $
Therefore $ \frac{x+y}{1-xy}=-z\Rightarrow x+y=-z+xyz $
Therefore $ x+y+z=xyz $ .
Dividing by xyz, we get $ \frac{1}{yz}+\frac{1}{xz}+\frac{1}{xy}=1 $ .
Note: Students should remember this question as a formula.
BETA
