Trigonometry & Inverse Trigonometry
Sum of cot inverse — Finding n
nta_pyq_2024_apr
Grade 12
Question:
For $n\in\mathbb{N}$, if $\cot^{-1}3+\cot^{-1}4+\cot^{-1}5+\cot^{-1}n=\dfrac{\pi}{4}$, then $n$ is equal to
Step-by-Step Solution
Key Concept: Convert to $\tan^{-1}$: $\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{4}+\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{n}=\frac{\pi}{4}$. Use the addition formula $\tan^{-1}a+\tan^{-1}b=\tan^{-1}\frac{a+b}{1-ab}$ repeatedly.
Combining: $\tan^{-1}(23/24)+\tan^{-1}(1/n)=\pi/4\Rightarrow 1/n=1/47\Rightarrow n=47$.
Correct Answer: 47