Answer:
Step 4 is incorrect ⇒ C
Step-by-step explanation:
Let us check step by step
∵ sin(x + y)
∵ sin(z) = cos([tex]\frac{\pi}{2}[/tex] - z)
→ Replace z by (x + y)
∴ sin(x + y) = cos([tex]\frac{\pi }{2}[/tex] - [x + y])
∴ Step 2 is correct
∵ cos([tex]\frac{\pi }{2}[/tex] - [x + y]) = cos([tex]\frac{\pi }{2}[/tex] - x - y])
∵ take [[tex]\frac{\pi }{2}[/tex] - x] as the 1st angle and y as the 2nd angle
∴ cos([tex]\frac{\pi }{2}[/tex] - x - y]) = cos([[tex]\frac{\pi }{2}[/tex] - x] - y)
∴ Step 3 is correct
∵ cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
→ Replace a by ([[tex]\frac{\pi }{2}[/tex] - x] and b by y
∴ cos([[tex]\frac{\pi }{2}[/tex] - x] - y) = cos([tex]\frac{\pi }{2}[/tex] - x)cos(y) + sin([tex]\frac{\pi }{2}[/tex] - x)sin(y)
∴ Step 4 is incorrect, the second angle is y not (-y)
∵ cos([tex]\frac{\pi }{2}[/tex] - x) = sin(x)
∵ sin([tex]\frac{\pi }{2}[/tex] - x) = cos(x)
∴ cos([tex]\frac{\pi }{2}[/tex] - x)cos(y) + sin([tex]\frac{\pi }{2}[/tex] - x)sin(y) = sin(x)cos(y) + cos(x)sin(y)
∴ Step 5 is correct
