Question

If sin $\alpha$ = $\frac{5}{13}$ and sin $\beta$ = $\frac{4}{5}$ then the value of cos $\alpha$ cos $\beta$ - sin $\alpha$ sin $\beta$ is

• A. $\frac{16}{65}$
• B. $\frac{9}{65}$
• C. $\frac{11}{65}$
• D. $\frac{15}{65}$

Answer 1

The correct option is A

$\frac{16}{65}$

5,12,13 are the sides of a right-angled $△$ and 3, 4, 5 are also sides of another right angled $△$ (Pythagorean triplets)

sin $\alpha$ = $\frac{5}{13}$;

$\therefore$ cos $\alpha$ = $\frac{12}{13}$

sin $\beta$ = $\frac{5}{13}$;

$\therefore$ cos $\beta$ = $\frac{3}{5}$

Substituting the values of cos $\alpha$, cos $\beta$, sin $\alpha$ and sin $\beta$ we get :

cos $\alpha$ cos $\beta$ - sin $\alpha$ sin $\beta$ = $(\frac{12}{13}$ x $\frac{3}{5}$ - $\frac{5}{13}$ x $\frac{4}{5})$ = $\frac{36}{65}-\frac{20}{65}$

= $\frac{16}{65}$