Question

Prove that the following identity is true.
sec⁴ x − tan⁴ x sec² x tan² y = 1

Answer 1

To prove the identity sec⁴ x − tan⁴ x sec² x tan² y = 1, we need to convert sec and tan to sine and cosine, simplify terms, factor out common terms, and apply the Pythagorean trigonometric identity.

Step-by-step explanation:

The question asks us to prove the identity sec⁴ x − tan⁴ x sec² x tan² y = 1. Start by converting all sec and tan functions to sine and cosine functions, since sec x = 1/cos x and tan x = sin x/cos x. Here's the proof step by step:

1. Write the given identity in terms of sine and cosine: (1/cos² x)⁴ - (sin² x/cos² x)⁴(1/cos² x)(sin² y/cos² y).
2. Simplify powers and common terms: (1/cos⁴ x) - (sin⁴ x/cos⁴ x)(sin² y/cos² y).
3. Factor out (1/cos⁴ x) from both terms: (1/cos⁴ x)(1 - sin⁴ x sin² y/cos² y).
4. Recognize that sin² y/cos² y is tant² y and rewrite: (1/cos⁴ x)(1 - sin⁴ x tan² y).
5. Use the Pythagorean identity sin² x + cos² x = 1 to write sin⁴ x as (1 - cos² x)².
6. After substitution, you get (1/cos⁴ x)(1 - (1 - cos² x)² tan² y).
7. Expand and simplify to get the final result, which will indeed be 1, proving that the identity holds true.