Question
If \alpha and \beta are two angles such that \sin \alpha +\cos \beta =\dfrac{3}{5}, \cos \alpha +\cos \beta =\dfrac{4}{5} , find the value of \cos \alpha \sin \beta
If \alpha and \beta are two angles such that \sin \alpha +\cos \beta =\dfrac{3}{5}, \cos \alpha +\cos \beta =\dfrac{4}{5} , find the value of \cos \alpha \sin \beta
Squaring both sides of two given equations would allow us to use Pythagorean Identity.
\because \sin \alpha +\cos \beta =\dfrac{3}{5}
Square the equation
\therefore (\sin \alpha +\cos \beta )^2
Square the equation
\therefore (\cos \alpha +\cos \beta)^2
Addition of (1) and (2), and using sum identity for sines function
\sin(\alpha +\beta) = -\dfrac{1}{2}
Subtraction (2) from (1) gives
\cos 2\beta -\cos 2\alpha +2\sin(\alpha -\beta) = -\dfrac{7}{25}
Applying the sum to product identity yields
\sin(\alpha -\beta) = -\dfrac{7}{25}
Using product to sum identity
\cos \alpha\sin \beta = \dfrac{1}{2} [\sin(\alpha+\beta ) - \sin(\alpha - \beta) ]
= \dfrac{1}{2}(- \dfrac{1}{2}+\dfrac{7}{25})
=-\dfrac{11}{100}