Inverse Trigonometric Functions Question 42
Question: If $ \alpha \in ( -\frac{3\pi }{2},-\pi ) $ then the value of $ {{\tan }^{-1}}(cot\alpha ) $ - $ {{\cot }^{-1}}(tan\alpha )+si{n^{-1}}(sin\alpha )+co{s^{-1}}(cos\alpha ) $ is equal to
Options:
A) $ 2\pi +a $
B) $ \pi +a $
C) 0
D) $ \pi -a $
Show Answer
Answer:
Correct Answer: C
For $ \alpha \in ( -\frac{3\pi }{2},-\pi ),\tan \alpha <0 $
$ \Rightarrow {{\tan }^{-1}}(cot\alpha )-co{t^{-1}}(tan\alpha )\times $
$ {{\tan }^{-1}}(cot\alpha )-[ \frac{\pi }{2}-{{\tan }^{-1}}(tan\alpha ) ] $
$ ={{\tan }^{-1}}(cot\alpha )+{{\tan }^{-1}}(tan\alpha )-\frac{\pi }{2} $
$ =-\pi $ Also for points in the second quadrant, we have $ {{\sin }^{-1}}(\sin \alpha )+{{\cos }^{-1}}(cos\alpha )=\pi $
BETA
