calculus

1. TRIGONOMETRIC IDENTITIES
ANGELO S. REYES

2. If and only if the measure of radius is 1 then:
𝑠𝑖𝑛𝑒 𝜃 = sin 𝜃 =
𝑦
𝑟
=
𝑦
1
= 𝑦
𝑐𝑜𝑠𝑖𝑛𝑒 𝜃 = cos 𝜃 =
𝑥
𝑟
=
𝑥
1
= 𝑥
sin 𝜃 = 𝑦
cos 𝜃 = 𝑥

3. Quotient Identities
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = tan 𝜃 =
𝑦
𝑥
=
sin 𝜃
cos 𝜃
𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = c𝑜𝑡 𝜃 =
𝑥
𝑦
=
cos 𝜃
sin 𝜃

4. Reciprocal Identities
𝑐𝑜𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = csc 𝜃 =
1
𝑦
=
1
sin 𝜃
𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = sec 𝜃 =
1
𝑥
=
1
cos 𝜃
𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = cot 𝜃 =
1
𝑦
𝑥
=
1
tan 𝜃

5. Pythagorean Identities
Since cos 𝜃 = 𝑥, sin 𝜃 = 𝑦, 𝑟 = 1
𝑦2 + 𝑥2 = 𝑟2
sin2
𝜃 + cos2
𝜃 = 1

6. Pythagorean Identities
Since tan 𝜃 =
𝑦
𝑥
, s𝑒𝑐 𝜃 =
1
𝑥
1 +
𝑦
𝑥
2
=
1
𝑥
2
1 + tan2 𝜃 = sec2 𝜃

7. Pythagorean Identities
Since cot 𝜃 =
𝑥
𝑦
, cs𝑐 𝜃 =
1
𝑦
1 +
𝑥
𝑦
2
=
1
𝑦
2
1 + cot2 𝜃 = csc2 𝜃

8. Summary of Pythagorean Identities
1 + tan2
𝜃 = sec2
𝜃 1 + cot2
𝜃 = csc2
𝜃 sin2
𝜃 + cos2
𝜃 = 1
sec2
𝜃 − 1 = tan2
𝜃
sec2
𝜃 − tan2
𝜃 = 1
1 − csc2
𝜃 = cot2
𝜃
𝑐𝑠𝑐2
𝜃 − c𝑜𝑡2
𝜃 = 1
1 − cos2
𝜃 = sin2
𝜃
1 − sin2
𝜃 = cos2
𝜃

9. Prove the following identities:
• cot2 𝜃 = 𝑐𝑜𝑡𝜃
1
tan 𝜃
• cos 𝜃 = sin 𝜃 cos 𝜃 csc 𝜃
• sin 𝜃 cos 𝜃 = sin2 𝜃 cos2 𝜃 csc 𝜃 sec 𝜃
• 1 = (sec 𝜃 − tan 𝜃)(sec 𝜃 + tan 𝜃)
• 2 cos2 𝜃 − 1 = 1 − 2 sin2 𝜃
• sin4
𝜃 + cos2
𝜃 sin2
𝜃 = sin2
𝜃
• csc 𝑥 sec 𝑥 =
csc 𝑥+sec 𝑥
cos 𝑥+sin 𝑥

11. Prove the following identities:
• sec 𝜃 csc 𝜃 = tan 𝜃 + cot 𝜃
• cot 𝑥 + csc 𝑥 sec 𝑥 =
sin 𝑥
1−cos 𝑥
+
tan 𝑥
1+cos 𝑥
• sin 𝜃 cos 𝜃 = sin2 𝜃 cos2 𝜃 csc 𝜃 sec 𝜃

13. Exercises:
• sin 𝜃 cot 𝜃 = cos 𝜃
• sin 𝜃 csc 𝜃 cos 𝜃 sec 𝜃 = 1
• tan 𝜃 cos 𝜃 csc 𝜃 = 1
• cot 𝜃 csc 𝜃 sin 𝜃 = cot 𝜃
• sec 𝜃 tan 𝜃 = sin 𝜃 sec2 𝜃
• sin2
𝜃 = (1 + cos 𝜃)(1 − cos 𝜃)
• sec 𝜃 − tan 𝜃 2 + 2 sec 𝜃 tan 𝜃 = sec2 𝜃 + tan2 𝜃
•
sin 𝜃
1−cos 𝜃
+
tan 𝜃
1+cos 𝜃
= csc 𝜃 sec 𝜃 + cot 𝜃

14. Sum and Difference Identities for Cosine
• cos 𝛼 − 𝛽 = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽 General Addition 1
• cos 𝛼 + 𝛽 = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽

15. Example 1: cos
𝜋
12

16. Example 2:
cos 100° cos 10° + sin 100° sin 10°

17. Example 3: Prove co𝑡 𝐴 cot 𝐵 − 1 =
cos 𝐴+𝐵
sin 𝐴 sin 𝐵

18. Let 𝐴 =
𝜋
2
, using General addition Formula1

19. Let 𝐵 =
𝜋
2
− 𝐶; 𝐴 =
𝜋
2

20. tan
𝜋
2
− 𝐵

21. sec
𝜋
2
− 𝐵

22. csc
𝜋
2
− 𝐵

23. cot
𝜋
2
− 𝐵

24. If and only if the measure of radius is 1 then:
𝑠𝑖𝑛𝑒 𝜃 = sin 𝜃 =
𝑦
𝑟
=
𝑦
1
= 𝑦
𝑐𝑜𝑠𝑖𝑛𝑒 𝜃 = cos 𝜃 =
𝑥
𝑟
=
𝑥
1
= 𝑥
sin 𝜃 = 𝑦
cos 𝜃 = 𝑥

25. Quotient Identities
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = tan 𝜃 =
𝑦
𝑥
=
sin 𝜃
cos 𝜃
𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = c𝑜𝑡 𝜃 =
𝑥
𝑦
=
cos 𝜃
sin 𝜃

26. Reciprocal Identities
𝑐𝑜𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = csc 𝜃 =
1
𝑦
=
1
sin 𝜃
𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = sec 𝜃 =
1
𝑥
=
1
cos 𝜃
𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = cot 𝜃 =
1
𝑦
𝑥
=
1
tan 𝜃

27. Summary of Pythagorean Identities
1 + tan2
𝜃 = sec2
𝜃 1 + cot2
𝜃 = csc2
𝜃 sin2
𝜃 + cos2
𝜃 = 1
sec2
𝜃 − 1 = tan2
𝜃
sec2
𝜃 − tan2
𝜃 = 1
1 − csc2
𝜃 = cot2
𝜃
𝑐𝑠𝑐2
𝜃 − c𝑜𝑡2
𝜃 = 1
1 − cos2
𝜃 = sin2
𝜃
1 − sin2
𝜃 = cos2
𝜃

28. Sum and Difference Identities for Cosines
• cos 𝛼 − 𝛽 = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽 General Addition 1
• cos 𝛼 + 𝛽 = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽

29. Sum and Difference Identities for Sines
• sin 𝛼 − 𝛽 = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽
• 𝑠𝑖𝑛 𝛼 + 𝛽 = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽

30. Exercises:
• sin(30° + 45°)
• cos(30° − 45°)
• sin(60° + 90°)
• cos(120° + 135°)
• cos(90° − 30°)
• sin
5𝜋
12
• cos
23𝜋
12
• sin
7𝜋
12