First, note that ctg(t) = 1/tan(t), so ctg(t) * sin(-t) can be rewritten as sin(-t) / tan(t). Since sin(-t) = -sin(t), and tan(t) = sin(t) / cos(t), we get:
Next, cos(2π - t) can be simplified using the difference of angles formula for cosine: cos(2π - t) = cos(2π)cos(t) + sin(2π)sin(t). Since cos(2π) = 1 and sin(2π) = 0, this simplifies to:
cos(2π - t) = cos(t)
Putting it all together, the expression simplifies to:
Let's simplify the given expression:
ctg(t) * sin(-t) + cos(2π - t)
First, note that ctg(t) = 1/tan(t), so ctg(t) * sin(-t) can be rewritten as sin(-t) / tan(t). Since sin(-t) = -sin(t), and tan(t) = sin(t) / cos(t), we get:
sin(-t) / tan(t) = -sin(t) / (sin(t) / cos(t)) = -cos(t)
Next, cos(2π - t) can be simplified using the difference of angles formula for cosine: cos(2π - t) = cos(2π)cos(t) + sin(2π)sin(t). Since cos(2π) = 1 and sin(2π) = 0, this simplifies to:
cos(2π - t) = cos(t)
Putting it all together, the expression simplifies to:
cos(t) + cos(t) = 0Therefore, ctg(t) * sin(-t) + cos(2π - t) simplifies to 0.