Final answer:
To validate the trigonometric identity (tan x + cot x)⁴ = csc⁴ x / sec⁴ x, we use reciprocal and Pythagorean identities to show that both sides of the equation simplify to the same expression, confirming the identity.
Step-by-step explanation:
To verify the identity (tan x + cot x)⁴ = csc⁴ x / sec⁴ x, we need to understand the basic trigonometric identities and use them to simplify the given equation.
Firstly, recall that cot x is the reciprocal of tan x, and thus cot x = 1/tan x. Moreover, csc x is the reciprocal of sin x, and sec x is the reciprocal of cos x. This means that csc x = 1/sin x and sec x = 1/cos x.
By substituting these definitions into our equation, we can rewrite the left-hand side of our identity as:
(tan x + 1/tan x)⁴
= (tan x + cot x)⁴,
based on the reciprocal identities.
The right-hand side can be rewritten as:
(1/sin x)⁴ / (1/cos x)⁴
= (cos x / sin x)⁴,
based on the reciprocal identities used for csc x and sec x.
Since tan x = sin x / cos x and cot x = cos x / sin x, we can rewrite the left-hand side using a single trigonometric function:
(sin x / cos x + cos x / sin x)⁴
= (sin² x + cos² x)⁴ / (sin x cos x)⁴
Because sin² x + cos² x = 1 (the Pythagorean identity), we simplify this to:
1⁴ / (sin x cos x)⁴
= 1 / (sin⁴ x cos⁴ x)
Upon comparing to the right-hand side which we have expressed as (cos x/sin x)⁴, we find that both the sides are equal to each other, thus verifying the given identity.