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Chapter 19: Problem 1849

\(\cot ^{-1} 1+\cot ^{-1} 3+\cot ^{-1} 5+\cot ^{-1} 7+\cot ^{-1} 8=\) (a) \((\pi / 4)\) (b) \((\pi / 2)\) (c) \((3 \pi / 4)\) (d) \((\pi / 3)\)

Short Answer

Expert verified
None of the given options match the result.

Step by step solution

01

Calculate inverse cotangents using equivalent inverse tangent values

Using the property \(\cot^{-1}(\tan(\alpha)) = \frac{\pi}{2} - \alpha\), we can rewrite the sum as follows: \[ \cot^{-1}(1) + \cot^{-1}(3) + \cot^{-1}(5) + \cot^{-1}(7) + \cot^{-1}(8) = (\frac{\pi}{2} - \tan^{-1}(1)) + (\frac{\pi}{2} - \tan^{-1}(3)) + (\frac{\pi}{2} - \tan^{-1}(5)) + (\frac{\pi}{2} - \tan^{-1}(7)) + (\frac{\pi}{2} - \tan^{-1}(8)) \]
02

Combine terms and rearrange the expression

Now, we will combine the terms containing \(\frac{\pi}{2}\) and those containing inverse tangent functions: \[ = 5\cdot \frac{\pi}{2} - (\tan^{-1}(1) + \tan^{-1}(3) + \tan^{-1}(5) + \tan^{-1}(7) + \tan^{-1}(8)) \]
03

Sum up the inverse tangent values

Add the values of the inverse tangent functions: \[ = 5\cdot \frac{\pi}{2} - (\frac{\pi}{4} + \tan^{-1}(3) + \tan^{-1}(5) + \tan^{-1}(7) + \tan^{-1}(8)) \] Notice that the angles of the inverse tangent functions do not correspond to any well-known values, which means that we can't simplify them further.
04

Compare the result with the given options

Looking at the given options, we can see that none of them match our result. Therefore, the exercise has no correct solution among the given alternatives.

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Most popular questions from this chapter

\({ }^{\infty} \sum_{r=1} \tan ^{-1}\left(1 / 2 r^{2}\right)=\) (a) \((\pi / 4)\) (b) \((\pi / 2)\) (c) \(\tan ^{-1}(\mathrm{n})-(\pi / 4)\) (d) \(\tan ^{-1}(n+1)-(\pi / 4)\)

The number of values of \(\theta\) in the interval \([0,2 \pi]\) satisfying the equation \(\tan 2 \theta \tan \theta=1\) is (a) 4 (b) 5 (c) 6 (d) 7

The value of \(\left.\operatorname{cosec}^{-1} \sqrt{5}+\operatorname{cosec}^{-1} \sqrt{(} 65\right)+\operatorname{cosec}^{-1} \sqrt{(325)}+\ldots+\infty\) is (a) \(\pi\) (b) \((3 \pi / 4)\) (c) \((\pi / 4)\) (d) \((\pi / 2)\)

If \(\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=[(2 \pi) / 3]\) then \(\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\) (a) \((\pi / 3)\) (b) \((5 \pi / 6)\) (c) \((\pi / 2)\) (d) \((3 \pi / 2)\)

\(\sec ^{2}\left(\tan ^{-1} 3\right)+\operatorname{cosec}^{2}\left(\tan ^{-1} 5\right)=\) (a) 276 (b) \([(276) / 25]\) (c) 36 (d) 6
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