Find the value of sin(2 tan⁻¹(1/3)) + cos(tan⁻¹ 2√2) | NCERT Class 12 Maths Ex 2.3 Q8

NCERT Class 12 Mathematics Solutions – Exercise 2.3 Question 8

NCERT Class 12 Maths | Chapter 2 Inverse Trigonometric Functions

Question

Find the value of \[ \sin \left( 2\tan^{-1}\frac{1}{3} \right) + \cos \left( \tan^{-1} 2\sqrt{2} \right) \]

[NCERT Ex 2.3, Question 8, Page 36]

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Solution

We have:

\[ \sin \left( 2\tan^{-1}\frac{1}{3} \right) + \cos \left( \tan^{-1} 2\sqrt{2} \right) \]

Using identities step by step:

\[ \sin\left(2\tan^{-1}\frac{1}{3}\right) = \sin \Biggl[ \sin^{-1} \Bigl( \frac{2 \cdot \frac{1}{3}}{1 + (\frac{1}{3})^2} \Bigr) \Biggr] \]

\[ \cos\left(\tan^{-1}2\sqrt{2}\right) = \cos\left(\cos^{-1}\frac{1}{3}\right) \]

\[ = \sin\Biggl[ \sin^{-1} \Bigl( \frac{\frac{2}{3}}{1 + \frac{1}{9}} \Bigr) \Biggr] + \frac{1}{3} \]

\[ = \sin\Biggl[ \sin^{-1} \Bigl( \frac{2 \cdot 9}{3 \cdot 10} \Bigr) \Biggr] + \frac{1}{3} = \sin\Biggl[ \sin^{-1} \frac{3}{5} \Biggr] + \frac{1}{3} \]

\[ = \frac{3}{5} + \frac{1}{3} = \frac{14}{15} \]