Question

If $secA=\frac{17}{8}$ 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

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

The correct options are
A $\frac{85}{36}$
B $\frac{85}{84}$
C $-\frac{85}{84}$
D $-\frac{85}{36}$

$\sec A=\frac{17}{8}\Rightarrow \cos A=\frac{8}{17}\Rightarrow \sin A=\pm \frac{15}{17}$
$\csc B=\frac{5}{4}\Rightarrow \sin B=\frac{4}{5}\Rightarrow \cos B=\pm \frac{3}{5}$
Now $\cos (A+B)=\cos A\cos B-\sin A\sin B$
$\Rightarrow \cos (A+B)=\frac{8}{17}.\frac{5}{4}-(\pm \frac{15}{17}.\pm \frac{3}{5})=\pm \frac{36}{85}$ or $\pm \frac{84}{85}$
$\therefore \sec (A+B)=\pm \frac{85}{36},\pm \frac{85}{84}$