Question

principle value of 2cot^-1(7)+cos^-1(3/5) is equal to ?

Answer 1

$$\begin{gathered} 2co{t}^{-1}7+{\cos }^{-1}\left(\frac{3}{5}\right) \\ If\theta ={\cos }^{-1}\left(\frac{3}{5}\right)then\cos \theta =\frac{3}{5}\Rightarrow cot\theta =\frac{3}{4} \\ Hence2co{t}^{-1}7+{\cos }^{-1}\left(\frac{3}{5}\right) \\ =2co{t}^{-1}7+co{t}^{-1}\frac{3}{4} \\ =co{t}^{-1}\left(\frac{7^2-1}{2\times 7}\right)+co{t}^{-1}\frac{3}{4} \\ =co{t}^{-1}\left(\frac{24}{7}\right)+co{t}^{-1}\left(\frac{3}{4}\right) \\ =co{t}^{-1}\left(\frac{{\displaystyle \frac{24}{7}}\times {\displaystyle \frac{3}{4}}-1}{{\displaystyle \frac{24}{7}}+{\displaystyle \frac{3}{4}}}\right)=co{t}^{-1}\left(\frac{{\displaystyle \frac{44}{28}}}{{\displaystyle \frac{117}{28}}}\right)=co{t}^{-1}\left(\frac{44}{117}\right)=69.39° \end{gathered}$$