Question

If $\sec A=\frac{17}{8}$ and $cosecB=\frac{5}{4}$ then $\sec (A+B)$ can have the value equal to

• A. $\frac{85}{36}$
• B. $-\frac{85}{36}$
• C. $-\frac{85}{84}$
• D. $\frac{85}{84}$

Answer 1

The correct option is B $-\frac{85}{36}$

$\cos A=\frac{8}{17}\Rightarrow \sin A=\frac{15}{17}$

$\sin B=\frac{4}{5}\Rightarrow \cos B=\frac{3}{5}$

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

$\Rightarrow \cos (A+B)=\frac{8}{17}\times \frac{3}{5}-\frac{15}{17}\times \frac{4}{5}$

$\Rightarrow \cos (A+B)=\frac{-36}{85}$

$\Rightarrow \sec (A+B)=-\frac{85}{36}$