Final answer:
The trigonometric identity cot²x - tan²x = cosec²x - sec²x can be verified by rewriting cot and tan in terms of sine and cosine, and simplifying both sides using these relations.
Step-by-step explanation:
The student has asked to verify the trigonometric identity cot²x - tan²x = cosec²x - sec²x. To do this, we'll rewrite each term using basic trigonometric identities and show that both sides of the equation are equal.
Starting with the left side of the identity:
- cot²x can be written as (cos²x/sin²x)
- tan²x can be expressed as (sin²x/cos²x)
Similarly, for the right side of the identity:
- cosec²x, which is 1/sin²x
- sec²x, which is 1/cos²x
Now, we rewrite the equation as:
(cos²x/sin²x) - (sin²x/cos²x) = (1/sin²x) - (1/cos²x)
With common denominators, we get:
((cos´x - sin´x) / (sin²x cos²x)) = ((cos²x - sin²x) / (sin²x cos²x))
Note that cos´x - sin´x can be factored into (cos²x + sin²x)(cos²x - sin²x), and since cos²x + sin²x = 1, we have:
(1)(cos²x - sin²x) / (sin²x cos²x)
Thus, proving the identity cot²x - tan²x = cosec²x - sec²x.