MCR 3U Grade 11 University Preparation
TRIGONOMETRIC IDENTITIES
Prove the following identities.
1. 2sin2x – 1 = sin2x – cos2x
2. (1 + sin x)(1 – sin x) = cos2x
3. (sin x + cos x)( sin x – cos x) = 1 – 2cos2x
4. cos2x = sin2x + 2cos2x – 1
5. cos3x + cosx sin2x = cos x
6. (1 – cos2x)(1 + cos2x) = 2sin2x – sin4x
7. (sin x + cos x)2 = 1 + 2 sin x ∙ cos x
8. (sin x + cos x)2 + (sin x – cos x)2 = 2
9. sin4x – cos4x = 2sin2x – 1
10. sin4x – cos4x = 1 – 2cos2x
11. sin4x + cos4x = 1 – 2 sin2x ∙ cos2x
𝟏 + 𝐜𝐨𝐬 𝒙
12. 𝟏 − 𝐜𝐨𝐬 𝒙 = 1 +
13.
𝟏
𝟏 − 𝐜𝐨𝐬 𝒙
𝟏
+
𝟐 𝐜𝐨𝐬 𝒙 (𝟏 + 𝐜𝐨𝐬 𝒙)
𝐬𝐢𝐧𝟐 𝒙
𝟏
𝟏 + 𝐜𝐨𝐬 𝒙
𝟏
=
𝟐
𝐬𝐢𝐧𝟐 𝒙
𝟏
14. 𝐬𝐢𝐧𝟐 𝒙 + 𝐜𝐨𝐬𝟐 𝒙 = 𝐬𝐢𝐧𝟐 𝒙 ∙ 𝐜𝐨𝐬𝟐 𝒙
15.
16.
𝟏 + 𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝐬𝐢𝐧 𝒙+ 𝐜𝐨𝐬 𝒙
𝟏 + 𝐬𝐢𝐧 𝜽 + 𝐜𝐨𝐬 𝜽
𝟏 – 𝐬𝐢𝐧 𝜽 + 𝐜𝐨𝐬 𝜽
= sin x + cos x
=
𝟏 + 𝐬𝐢𝐧 𝜽
𝐜𝐨𝐬 𝜽
17.
𝐬𝐢𝐧𝟑 𝒙 + 𝐜𝐨𝐬 𝟑 𝒙
𝐬𝐢𝐧 𝒙 + 𝐜𝐨𝐬 𝒙
= 1 – sin x cos x
18. cos x × tan x = sin x
19. sin x × cot x = cos x
20. tan x = tan2 x × cot x
21. 1 – sin x cos x tan x = cos2 x
22. 1 + tan x =
𝐬𝐢𝐧 𝒙 + 𝐜𝐨𝐬 𝒙
𝐜𝐨𝐬 𝒙
𝟏
𝟏
23. (𝐜𝐨𝐬 𝒙 + 𝟏) (𝐜𝐨𝐬 𝒙 − 𝟏) = tan2 x
24.
25.
26.
27.
28.
29.
𝟏
𝐜𝐨𝐬𝟐 𝒙
– tan2 x = 1
𝐜𝐨𝐬 𝒙
𝟏
=
– tan x
𝐬𝐢𝐧 𝒙 + 𝟏 𝐜𝐨𝐬 𝒙
𝟏 + 𝐬𝐢𝐧 𝒙
2
×
tan
x=
𝟏 − 𝐜𝐨𝐬 𝒙
𝟏 + 𝐜𝐨𝐬 𝒙
𝟏 – 𝐬𝐢𝐧 𝒙
𝐜𝐨𝐭 𝒙
𝐭𝐚𝐧 𝒙
=
× cot2x =
𝟏 + 𝐜𝐨𝐬 𝒙
𝟏 – 𝐬𝐢𝐧 𝒙
𝟏 + 𝐬𝐢𝐧 𝒙
𝟏 –𝐜𝐨𝐬 𝒙
𝟏 − 𝐬𝐢𝐧𝟐 𝒙
𝟏 − 𝐜𝐨𝐬𝟐 𝒙
𝐬𝐢𝐧 𝒙 + 𝐭𝐚𝐧 𝒙
𝟏 + 𝐜𝐨𝐬 𝒙
= tan x
30. tan2x – sin2x = sin2x × tan2x
31. sin x + tan x = tan x ∙ (1 + cos x)
𝐭𝐚𝐧 𝒙 𝐬𝐢𝐧 𝒙
32. 𝐭𝐚𝐧 𝒙 + 𝐬𝐢𝐧 𝒙 =
𝐭𝐚𝐧 𝒙 − 𝐬𝐢𝐧 𝒙
𝐭𝐚𝐧 𝒙 𝐬𝐢𝐧 𝒙
𝐬𝐞𝐜 𝜽
33. tan 𝜽 = 𝐜𝐬𝐜 𝜽
𝟏
𝟏
𝟏
𝟏
34. 2csc2x = 𝟏 − 𝐜𝐨𝐬 𝒙 + 𝟏 + 𝐜𝐨𝐬 𝒙
35. 2sec2x = 𝟏 − 𝐬𝐢𝐧 𝒙 + 𝟏 + 𝐬𝐢𝐧 𝒙
36. tan x cot x – sec x cos x
37.
𝟏 + 𝐜𝐬𝐜 𝒙
𝐬𝐞𝐜 𝒙
𝐬𝐞𝐜 𝒙 + 𝟏
= cos x + tan x
𝐜𝐨𝐬 𝒙 + 𝟏
38. 𝐬𝐞𝐜 𝒙 − 𝟏 + 𝐜𝐨𝐬 𝒙 − 𝟏 = 0
39. sec2 x + csc2 x = sec2x ∙ csc2x
40. tan x + cot x = sec x ∙ csc x
𝟏
41. 𝟏 + 𝐜𝐨𝐬 𝒙 = csc2 x – csc x ∙ cot x
𝐜𝐨𝐬 𝒙
42. 𝐬𝐞𝐜 𝒙 −
𝐬𝐢𝐧 𝒙
=
𝐜𝐨𝐭 𝒙
𝐜𝐬𝐜 𝒙 + 𝐜𝐨𝐭 𝒙
43. 𝐜𝐬𝐜 𝒙 − 𝐜𝐨𝐭 𝒙 =
𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙 − 𝐭𝐚𝐧 𝒙
𝐜𝐬𝐜 𝒙
𝟏 + 𝟐 𝐜𝐨𝐬 𝒙 + 𝐜𝐨𝐬𝟐 𝒙
𝒔𝒊𝒏𝟐 𝒙