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Sample Problems
Prove each of the following identities.
- tan x sin x + cos x = sec x
tan x
sin x cos x
- sin x sin x cos^2 x = sin^3 x
cos 1 + sin
1 + sin cos = 2 sec
cos x 1 sin x
cos x 1 + sin x
= 2 tan x
- cos^2 x = csc x cos x tan x + cot x
sin^4 x cos^4 x sin^2 x cos^2 x
tan^2 x tan^2 x + 1 = sin^2 x
1 sin x cos x
cos x 1 + sin x
- 1 2 cos^2 x = tan^2 x 1 tan^2 x + 1
- tan^2 = csc^2 tan^2 1
- sec x + tan x = cos x 1 sin x
csc sin
cot tan
- sin^4 x cos^4 x = 1 2 cos^2 x
- (sin x cos x)^2 + (sin x + cos x)^2 = 2
sin^2 x + 4 sin x + 3 cos^2 x
3 + sin x 1 sin x
cos x 1 sin x tan x = sec x
- tan^2 x + 1 + tan x sec x = 1 + sin x cos^2 x
Practice Problems
Prove each of the following identities.
- tan x + cos x 1 + sin x
cos x
- tan^2 x + 1 = sec^2 x
1 sin x
1 + sin x = 2 tan x sec x
- tan x + cot x = sec x csc x
1 + tan^2 x 1 tan^2 x
cos^2 x sin^2 x
- tan^2 x sin^2 x = tan^2 x sin^2 x
1 cos x sin x
sin x 1 cos x = 2 csc x
sec x 1 sec x + 1
1 cos x 1 + cos x
- 1 + cot^2 x = csc^2 x
csc^2 x 1 csc^2 x = cos^2 x
cot x 1 cot x + 1
1 tan x 1 + tan x
- (sin x + cos x) (tan x + cot x) = sec x + csc x
sin^3 x + cos^3 x sin x + cos x = 1 sin x cos x
cos x + 1 sin^3 x
csc x 1 cos x
1 + sin x 1 sin x
1 sin x 1 + sin x = 4 tan x sec x
- csc^4 x cot^4 x = csc^2 x + cot^2 x
sin^2 x cos^2 x + 3 cos x + 2
1 cos x 2 + cos x
tan x + tan y cot x + cot y = tan x tan y
1 + tan x 1 tan x
cos x + sin x cos x sin x
- (sin x tan x) (cos x cot x) = (sin x 1) (cos x 1)
- cos^2 x =
csc x cos x tan x + cot x Solution: We will start with the right-hand side. We will re-write everything in terms of sin x and cos x and simplify. We will again run into the Pythagorean identity, sin^2 x + cos^2 x = 1.
RHS =
csc x cos x tan x + cot x
sin x
cos x sin x cos x
cos x sin x
sin x
cos x 1 sin^2 x sin x cos x
cos^2 x sin x cos x
cos x sin x sin^2 x + cos^2 x sin x cos x
cos x sin x 1 sin x cos x = cos x sin x
cos x sin x 1
cos^2 x 1 = cos^2 x = LHS
sin^4 x cos^4 x sin^2 x cos^2 x
Solution: We can factor the numerator via the di§erence of squares theorem.
LHS =
sin^4 x cos^4 x sin^2 x cos^2 x
sin^2 x
(cos^2 x)^2 sin^2 x cos^2 x
sin^2 x + cos^2 x
sin^2 x cos^2 x
sin^2 x cos^2 x = sin^2 x + cos^2 x = 1 = RHS
tan^2 x tan^2 x + 1 = sin^2 x
Solution:
LHS =
tan^2 x tan^2 x + 1
sin x cos x
sin x cos x
sin^2 x cos^2 x sin^2 x cos^2 x
sin^2 x cos^2 x sin^2 x cos^2 x
cos^2 x cos^2 x
sin^2 x cos^2 x sin^2 x + cos^2 x cos^2 x
sin^2 x cos^2 x 1 cos^2 x
sin^2 x cos^2 x
cos^2 x 1 = sin^2 x = RHS
1 sin x cos x
cos x 1 + sin x Solution:
LHS = 1 sin x cos x
1 sin x cos x
1 sin x cos x
1 + sin x 1 + sin x
(1 sin x) (1 + sin x) cos x (1 + sin x)
1 sin^2 x cos x (1 + sin x)
= cos^2 x cos x (1 + sin x)
cos x 1 + sin x
= RHS
- 1 2 cos^2 x = tan^2 x 1 tan^2 x + 1 Solution:
RHS =
tan^2 x 1 tan^2 x + 1
sin^2 x cos^2 x
sin^2 x cos^2 x
sin^2 x cos^2 x
cos^2 x cos^2 x sin^2 x cos^2 x
cos^2 x cos^2 x
sin^2 x cos^2 x cos^2 x sin^2 x + cos^2 x cos^2 x = sin^2 x cos^2 x cos^2 x
cos^2 x sin^2 x + cos^2 x
sin^2 x cos^2 x sin^2 x + cos^2 x
sin^2 x cos^2 x 1 = sin^2 x cos^2 x =
1 cos^2 x
cos^2 x = 1 2 cos^2 x = LHS
- tan^2 = csc^2 tan^2 1
RHS = csc^2 tan^2 1 =
sin^2
sin cos
sin^2
sin^2 cos^2
cos^2
cos^2
cos^2 cos^2
1 cos^2 cos^2
sin^2 cos^2
sin cos
= tan^2 = LHS
- sec x + tan x = cos x 1 sin x Solution:
RHS = cos x 1 sin x
cos x 1 sin x
cos x 1 sin x
1 + sin x 1 + sin x
cos x (1 + sin x) (1 sin x) (1 + sin x) = cos x (1 + sin x) 1 sin^2 x
cos x (1 + sin x) cos^2 x
1 + sin x cos x
cos x
sin x cos x
= LHS
csc sin
cot tan
Solution: We will start with the left-hand side. We will re-write everything in terms of sin and cos and simplify. We will again run into the Pythagorean identity, sin^2 x + cos^2 x = 1 for all angles x.
LHS =
csc sin
cot tan
sin sin 1
cos sin sin cos
sin
sin
cos sin
cos sin
sin^2
cos^2 sin^2
1 cos^2 sin^2
sin^2 + cos^2
cos^2 sin^2
sin^2 sin^2
= 1 = RHS
- sin^4 x cos^4 x = 1 2 cos^2 x
Solution: LHS = sin^4 x cos^4 x =
sin^2 x
cos^2 x
sin^2 x + cos^2 x
sin^2 x cos^2 x
sin^2 x cos^2 x
1 cos^2 x
cos^2 x = 1 2 cos^2 x = RHS
