Help! Trig. Explain !

Question

Simplify

Csc^4x - cot^4x?

Answer 1

csc⁴x - cot⁴x

= (csc²x)² - (cot²x)²

Mathematical formula: (a² - b²) = (a + b)(a - b)

= (csc²x + cot²x)(csc²x - cot²x)

Pythagorean Identities: csc²θ - cot²θ = 1

= (csc²x + cot²x)(1)

Reciprocal identities: cscθ =1/sinθ and Quotient Identities: cotθ =cosθ/sinθ.

= 1/sin²x + cos²x/sin²x

Rewrite the expression with common denominator.

= [1 + cos²x] / sin²x.