using trig identities to write expressions in terms of a single trigonometric function or constant

Question

how to write these equations in terms of a single trigonometirc fucntion or constant

1.(cscx-1)(cscx+1) tanx

2.cos x sec²x cotx

Answer 1

1.(cscx-1)(cscx+1) tanx FOIL out the terms in parenthesis

(csc²x-csc x+cscx-1)(tan x) Simplify

(csc²x-1)(tan x) Using the Pythagorean identity, csc²x-1 = tan²x, substitute

(tan²x)(tan x) = tan³x

2.cos x sec²x cot x Substitute in sec x = 1/cos x and cot x = cos x/sin x

cos x (1/cos²x) (cos x/sin x) Simplify all the cos x

1/sin x = csc x