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CBSE Class 12 Math 2013 Solved Paper

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Question : 1 of 29
Marks: +1, -0
Write the principal value of tan⁡−1\tan^{-1} (1) + cos⁡−1(−12)\cos^{-1}\left(-\frac{1}{2}\right)
Solution:  
Let tan⁡−1\tan^{-1} (1) = y
⇒ tan y = 1 = tan (π4)\left(\frac{\pi}{4}\right) ⇒ y = π4\frac{\pi}{4}
⇒ tan⁡−1(1)\tan^{-1}(1) = π4\frac{\pi}{4}
cos⁡−1(−12)\cos^{-1}\left(-\frac{1}{2}\right) = z
⇒ cos z = −12-\frac{1}{2} = - cos π3\frac{\pi}{3} = cos (π−π3)\left(\pi - \frac{\pi}{3}\right) = cos (2π3)\left(\frac{2\pi}{3}\right)
⇒ z = 2π3\frac{2\pi}{3} ⇒ cos⁡−1(−12)\cos^{-1}\left(-\frac{1}{2}\right) = 2π3\frac{2\pi}{3}
∴ tan⁡−1\tan^{-1} (1) + cos⁡−1(−12)\cos^{-1}\left(-\frac{1}{2}\right) = π4+2π3\frac{\pi}{4} + \frac{2\pi}{3} = 11π12\frac{11\pi}{12}
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