SIMPLIFYING WITH TRIGONOMETRIC IDENTITIES

Simplify.

Problem 1 :

tan x(csc x - sin x)

Solution :

tan x(csc x - sin x)

Problem 2 :

(csc x - 1) (csc x + 1)

Solution :

(csc x - 1) (csc x + 1)

= (csc² x - 1) 

= cot²x

Problem 3 :

(sec x - tan x) (sec x + tan x)

Solution :

(sec x - tan x) (sec x + tan x)

= sec² x - tan² x

= 1

Problem 4 :

(1 + tan²x) (1 - sin²x)

Solution :

(1 + tan²x) (1 - sin²x)

= sec²x ⋅ cos²x

= (1/cosx)² ⋅ cos²x

= (1/cos²x) ⋅ cos²x

= 1

(1 + tan²x) (1 - sin²x) = 1

Problem 5 :

cos x(csc x - secx) - cot x

Solution :

cos x(csc x - secx) - cot x

Problem 6 :

sin² x(cot²x + 1)

Solution :

sin² x(cot²x + 1)

= sin² x(csc²x)

= sin²x(1/sin x)²

= sin²x(1/sin²x)

= 1

Problem 7 :

sin²x + sin²x cot²x

Solution :

sin²x + sin²x cot²x

= sin²x[1 + cot²x]

= sin²x ⋅ csc²x

= sin²x(1/sinx)²

= sin²x ⋅ (1/sin²x)

= 1

Problem 8 :

cot x sec x sin x

Solution :

cot x sec x sin x

= (cos x/sin x) ⋅ sec x ⋅ sin x

= (cos x/sin x) ⋅ (1/cos x) ⋅ sin x

= 1

Problem 9 :

Solution :