Final answer:
The trigonometric identity csc(x) - sin(x) simplifies to cos(x) × cot(x) by using the definition of csc(x), finding a common denominator, and applying the Pythagorean identity.
Step-by-step explanation:
To solve the trigonometric identity csc(x) - sin(x), we use the definition of the cosecant function, which is csc(x) = 1/sin(x). Let's rewrite the expression using this definition:
csc(x) - sin(x) = 1/sin(x) - sin(x)
To combine these terms, we need a common denominator, which in this case is sin(x). Multiplying the second term by sin(x)/sin(x) allows us to combine the terms:
(1 - sin2(x)) / sin(x)
We recognize 1 - sin2(x) as a Pythagorean identity, which is equivalent to cos2(x), hence:
cos2(x) / sin(x)
This can be further simplified by breaking up cos2(x) as cos(x) × cos(x), giving us:
cos(x) × (cos(x) / sin(x))
The term cos(x) / sin(x) is known as cotangent, cot(x). Thus, our final simplified identity is:
cos(x) × cot(x)