Question

# Write true or false.If $\sin x = \cos y$ then $x + y = {45^ \circ }$

Hint: Try to remember a principal value of x and see the complimentary value of y then add them and see whether we are getting ${45^ \circ }$ or not.
Let us do it by taking some examples first we know that $\sin \dfrac{\pi }{3} = \dfrac{{\sqrt 3 }}{2}$ and the same value of cosine exists when $\cos \dfrac{\pi }{6} = \dfrac{{\sqrt 3 }}{2}$
So in this case $\sin \dfrac{\pi }{3} = \cos \dfrac{\pi }{6} = \dfrac{{\sqrt 3 }}{2}$
$\begin{array}{l} \therefore x + y\\ = \dfrac{\pi }{3} + \dfrac{\pi }{6}\\ = \dfrac{{2\pi + \pi }}{6}\\ = \dfrac{{3\pi }}{6}\\ = \dfrac{\pi }{2} \end{array}$
And we also know that $\dfrac{\pi }{2} = {90^ \circ }$ so from here we can clearly see that the statement written is false
Note: we also know that $\sin x = \cos (90 - x)$ so this also settles it because we can also write $\cos y = \sin (90 - y)$ therefore $\sin x = \sin (90 - y)$ and now if we remove sin from both the sides we are left with $x = {90^ \circ } - y$ i.e., $x + y = {90^ \circ }$