The answer is \(\frac{7}{5}\)
Complimentary angles: the angles added up equal 90 degrees
When you have complimentary angles, the sine of one equals the cosine of the other and vice verse
Q and P are complimentary
sin(Q)=cos(P)
cos(Q)=sin(P)
Since you know that sin(Q)=\(\frac{4}{5}\) , that also means that cos(P)=\(\frac{4}{5}\)
Since sin(Q) is \(\frac{leg}{hypotenuse}\) , you can use the Pythagorean Theorem to find the length of the last leg.
\(a^{2} + b^{2} = c^{2}\)
\(a^{2} + (4)^{2} = (5)^{2}\)
\(a^{2} + 16 = 25\)
\(a^{2} = 9\)
\(a = 3\)
(It's a 3,4,5 triangle)
Now you have the length of the other leg so cos(Q)=\(\frac{3}{5}\)
cos(P)+cos(Q)
=\(\frac{4}{5} + \frac{3}{5}\)
=\(\frac{7}{5}\)
The answer is \(\frac{7}{5}\)
