verify the identity...
cot(-a)cos(-a)+sin(-a)=-csc(a)
To verify the given identity:
cot(-a)cos(-a) + sin(-a) = -csc(a)
We will start with the left side of the equation:
cot(-a) = cos(-a) / sin(-a) (Definition of cotangent)
cos(-a) = cos(a) (Cosine function is an even function)
sin(-a) = -sin(a) (Sine function is an odd function)
Substituting these values into the equation, we have:
cos(a) / -sin(a) * cos(a) + (-sin(a))
Now, let's simplify this expression:
cos(a) * cos(a) / -sin(a) - sin(a) (Multiplying the terms)
cos^2(a) / -sin(a) - sin(a) (Simplifying cos(a) * cos(a) to cos^2(a))
To continue simplifying, let's express cos^2(a) in terms of sin(a) using the identity:
cos^2(a) = 1 - sin^2(a) (Pythagorean identity)
Substituting this identity into the expression, we get:
(1 - sin^2(a)) / -sin(a) - sin(a)
Now, let's simplify further:
(1/sin(a) - sin^2(a)/sin(a)) - sin(a) (Distributing the negative sign)
1/sin(a) - sin(a) - sin(a) (Canceling out sin^2(a) / sin(a))
1/sin(a) - 2sin(a) (Combining like terms)
Finally, let's convert the right side of the equation to match the left side:
-csc(a) = -1/sin(a) (Reciprocal identity for cosecant)
Now, we can see that the left-hand side (-csc(a)) matches the simplified expression on the right-hand side (1/sin(a) - 2sin(a)).
Thus, the given identity cot(-a)cos(-a) + sin(-a) = -csc(a) is verified.