Final answer:
To verify the identity cot⁴ x - csc⁴ x = -2cot² x - 1, we express cot and csc in terms of sine and cosine, simplify, and factor to show that the identity holds true.
Step-by-step explanation:
The student is asking to verify the trigonometric identity cot⁴ x - csc⁴ x = -2cot² x - 1. The way to approach this is by expressing the cotangent and cosecant functions in terms of sine and cosine and then simplifying.
Firstly, recall that cot x = 1/tan x = cos x/sin x and csc x = 1/sin x. Using these definitions, we can rewrite cot⁴ x as (cos⁴ x)/(sin⁴ x) and csc⁴ x as 1/(sin⁴ x).
Now, the expression becomes ((cos⁴ x)/(sin⁴ x)) - 1/(sin⁴ x). Factoring out 1/(sin⁴ x) yields (cos⁴ x - 1)/(sin⁴ x). Since cos² x = 1 - sin² x, it follows that cos⁴ x = (cos² x)² = (1 - sin² x)². Substituting this into our expression we get [(1 - sin² x)² - 1]/(sin⁴ x).
Expanding the term in the numerator we have 1 - 2sin²x + sin⁴x - 1, which simplifies to -2sin²x + sin⁴x. Dividing by sin⁴x gives -2/sin²x + 1. Recognizing that sin² x = 1/csc² x, we can substitute and obtain -2cot² x + 1.
Lastly, factoring out a negative sign gives us -1(2cot² x + 1), which matches the right side of the original identity.