Question

Verify the identity algebraically: ( 1/sin(x) - 1/csc(x) = csc(x) - sin(x) )

a) True

b) False

Answer 1

To verify the given identity algebraically, we manipulate the left-hand side and simplify it to match the right-hand side, which confirms that the identity is true.

Step-by-step explanation:

To verify the identity algebraically, we need to manipulate the left-hand side of the equation and simplify it to see if it is equal to the right-hand side.

1. Start with the left-hand side of the equation: 1/sin(x) - 1/csc(x).
2. Convert both terms to have the same denominator: (csc(x) - sin(x))/(sin(x)*csc(x)).
3. Simplify the numerator: (csc(x) - sin(x))/(sin(x)*csc(x)) = (1 - sin(x))/sin(x).
4. Simplify the denominator: (csc(x) - sin(x))/(sin(x)*csc(x)) = 1.
5. The left-hand side simplifies to 1, which is equal to the right-hand side, csc(x) - sin(x).

Therefore, the identity is verified and the answer is True.