Some examples of using reciprocal, quotient, and Pythagorean identities to prove trigonometric identities.

1. TRIGONOMETRY Proving Trigonometric Identities

2. REVIEW Quotient Identities Reciprocal Identities Pythagorean Identities

3. Let’s start by working on the left side of the equation….

4. Rewrite the terms inside the second parenthesis by using the quotient identities

5. Simplify

6. To add the fractions inside the parenthesis, you must multiply by one to get common denominators

7. Now that you have the common denominators, add the numerators

8. Simplify

9. Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!

10. On to the next problem….

11. Let’s start by working on the left side of the equation….

12. We’ll factor the terms using the difference of two perfect squares technique

13. Using the Pythagorean Identities the second set of parenthesis can be simplified

14. Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!

15. On to the next problem….

16. Let’s start by working on the right side of the equation….

17. Multiply by 1 in the form of the conjugate of the denominator

18. Now, let’s distribute in the numerator….

19. … and simplify the denominator

20. ‘ Split’ the fraction and simplify

21. Use the Quotient and Reciprocal Identities to rewrite the fractions

22. And then by using the commutative property of addition…

23. … you’ve successfully proven the identity!

24. One more….

25. Let’s work on the left side of the equation…

26. Multiply each fraction by one to get the LCD

27. Now that the fractions have a common denominator, you can add the numerators

28. Simplify the numerator

29. Use the Pythagorean Identity to rewrite the denominator

30. Multiply the fraction by the constant

31. Use the Reciprocal Identity to rewrite the fraction to equal the expression on the right side of the equation