Question

If the numerical value of $tan\left(co{s}^{-1}\right(4/5)+ta{n}^{-1}(2/3\left)\right)$ is a/b then

• A. $a+b=23$
• B. $a-b=11$
• C. $3b=a+1$
• D. $2a=3b$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $a+b=23$
B $a-b=11$
C $3b=a+1$

Let $co{s}^{-1}\left(4/5\right)=\alpha$, that is, $cos\alpha =4/5$, so that
$tan\alpha =\sqrt{{\left(\frac{5}{4}\right)}^2-1}=\frac{3}{4}$ $(\because 0\alpha <\pi andcos\alpha >0)$
and $tan(co{s}^{-1}\frac{4}{5}+ta{n}^{-1}\frac{2}{3})=\frac{tan\alpha +\frac{2}{3}}{1-tan\alpha \frac{2}{3}}$
$=\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{2}{3}\cdot \frac{3}{4}}=\frac{17}{6}=\frac{a}{b}$ (Given)
So $a=17,b=6,a+b=23,a-b=11$ and $3b=a+1$