how do you simplify
cos(A+B)*cos B=sin(A+B)*sin B
can you show me the steps?
thank you very much!!!
To simplify the given equation, cos(A+B)*cos B = sin(A+B)*sin B, we can use the trigonometric identities. Here are the steps:
Step 1: Expand the left side of the equation using the cosine double-angle formula: cos(A+B) = cos A*cos B - sin A*sin B.
The equation becomes:
(cos A*cos B - sin A*sin B)*cos B = sin(A+B)*sin B
Step 2: Simplify the expression on the left side of the equation by distributing cos B:
cos A*cos B*cos B - sin A*sin B*cos B = sin(A+B)*sin B
Step 3: Use the identity sin(A+B) = sin A*cos B + cos A*sin B to rewrite the right side of the equation:
cos A*cos B*cos B - sin A*sin B*cos B = (sin A*cos B + cos A*sin B)*sin B
Step 4: Simplify both sides of the equation by distributing:
cos A*cos B*cos B - sin A*sin B*cos B = sin A*cos B*sin B + cos A*sin B*sin B
Step 5: Combine like terms on both sides of the equation:
cos A*cos B*cos B - sin A*sin B*cos B = sin A*cos B*sin B + cos A*sin B*sin B
Step 6: Factor out the common factor on both sides of the equation:
cos B(cos A*cos B - sin A*sin B) = sin B(sin A*cos B + cos A*sin B)
Step 7: Divide both sides of the equation by cos B * sin B:
cos A*cos B - sin A*sin B = sin A*cos B + cos A*sin B
Step 8: Rearrange the terms on both sides of the equation:
cos A*cos B - sin A*cos B = sin A*sin B + cos A*sin B
Step 9: Factor out the common factor on both sides of the equation:
(cos A - sin A)*cos B = (sin A + cos A)*sin B
Step 10: Divide both sides of the equation by (sin A + cos A):
cos B = sin B
Therefore, the simplified equation is cos B = sin B.