Find the value of the expression $\sin \bigg(2\tan^{-1}{\frac{1}{3}}\bigg)+\cos \big(\tan^{-1}2\sqrt 2\big)$

$\begin{array}{1 1} \frac{15}{14} \\ \frac{-15}{14} \\ \frac{14}{15} \\ \frac{-14}{15} \end{array}$

Toolbox:
• $$2tan^{-1}x=tan^{-1}\large\frac{2x}{1-x^2}$$
• $$tan^{-1}x=sin^{-1}\large\frac{x}{\sqrt{1+x^2}}$$
• $$tan^{-1}x=cos^{-1}\large\frac{1}{\sqrt{1+x^2}}$$
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By taking x=$$\large\frac{1}{3},\:\large\frac{2x}{1-x^2}=\large\frac{2.\large\frac{1}{3}}{1-\large\frac{1}{9}}=\large\frac{2}{3}.\large\frac{9}{8}=\large\frac{3}{4}$$

Substituting in the formula of $$2tan^{-1}x$$ we get
$$2tan^{-1}\large\frac{1}{3}=tan^{-1}\large\frac{3}{4}$$

By taking x=$$\large\frac{3}{4},\:\large\frac{x}{\sqrt{1+x^2}}=\large\frac{\large\frac{3}{4}}{\sqrt{1+\large\frac{9}{16}}}=\large\frac{3}{4}.\large\frac{4}{5}=\large\frac{3}{5}$$

Substituting in the above formula of $$tan^{-1}x$$ we get
$$tan^{-1}\large\frac{3}{4}=sin^{-1}\large\frac{3}{5}$$

Similarly by taking$$x=2\sqrt 2,\:\large\frac{1}{\sqrt{1+x^2}}=\large\frac{1}{\sqrt{1+8}}=\large\frac{1}{3}$$

Substituting in the above formula of $$tan^{-1}x$$ we get
$$tan^{-1} 2\sqrt 2=cos^{-1}\large\frac{1}{3}$$

Substituting tthe values in the given expression we get
$$sin \bigg( tan^{-1}\large\frac{3}{4} \bigg) +cos (tan^{-1} 2\sqrt 2)$$
$$= sin \bigg( sin^{-1}\large\frac{3}{5} \bigg)+cos \bigg(cos^{-1}\large\frac{1}{3} \bigg)$$
$$= \large\frac{3}{5}+\frac{1}{3}=\large\frac{14}{15}$$

edited Mar 15, 2013