Question

Упростить: Cos(a+3π/2)cos(a-5π/2)+sin(a-3π)cos(a+7π/2)

Answer 1

о формулам приведения:  

sin(a-3π/2)=-sin(3π/2-а) =cos a  

cos(2π-a) = cos a  

sin(π-a)=sin a  

cos(π/2+a)=-sin a  

 

cos a*cos a-sin a*(-sin a)=cos^2 a + sin^2 a =1  

cos^2 a + sin^2 a = 1 - основное тригонометрическое тождество.

Answer 2

To simplify the given expression, we'll use trigonometric identities to rewrite the trigonometric functions in a more manageable form. Here's the step-by-step solution:

Step 1: Convert the angles to their equivalent angles within the range [0, 2π):

• Cos(a + 3π/2) = Cos(a + π/2) = -Sin(a)
• Cos(a - 5π/2) = Cos(a - π/2) = Sin(a)
• Sin(a - 3π) = -Sin(a)
• Cos(a + 7π/2) = Cos(a + 3π/2) = -Sin(a)

Step 2: Substitute the equivalent angles into the expression:

Cos(a + 3π/2) * Cos(a - 5π/2) + Sin(a - 3π) * Cos(a + 7π/2) = (-Sin(a)) * (Sin(a)) + (-Sin(a)) * (-Sin(a)) = -Sin(a)^2 + Sin(a)^2 = 0

So, the simplified expression is 0.