If 0 2  x  S and secx = cosecy, then the value of sin (x+y) is :

secx = cosecy ⇒ cosx = siny $Rightarrow sin left(frac{pi}{2}-x ight)=sin y$ $Rightarrow y=frac{pi}{2}-x Rightarrow x+y=frac{pi}{2}$ $herefore sin (x+y)=sin frac{pi}{2}=1$

Alternative:

Let

x = y = 45°

∴ sec x = cosec y

sec 45° = cosec 45° = 2

sin (x + y) = sin (45° + 45°)

= sin 90° = 1