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# Proving Trigonometric Identities

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

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### Proving Trigonometric Identities

1. 1. TRIGONOMETRY Proving Trigonometric Identities
2. 2. REVIEW Quotient Identities Reciprocal Identities Pythagorean Identities
3. 3. Let’s start by working on the left side of the equation….
4. 4. Rewrite the terms inside the second parenthesis by using the quotient identities
5. 5. Simplify
6. 6. To add the fractions inside the parenthesis, you must multiply by one to get common denominators
7. 7. Now that you have the common denominators, add the numerators
8. 8. Simplify
9. 9. Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!
10. 10. On to the next problem….
11. 11. Let’s start by working on the left side of the equation….
12. 12. We’ll factor the terms using the difference of two perfect squares technique
13. 13. Using the Pythagorean Identities the second set of parenthesis can be simplified
14. 14. Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!
15. 15. On to the next problem….
16. 16. Let’s start by working on the right side of the equation….
17. 17. Multiply by 1 in the form of the conjugate of the denominator
18. 18. Now, let’s distribute in the numerator….
19. 19. … and simplify the denominator
20. 20. ‘ Split’ the fraction and simplify
21. 21. Use the Quotient and Reciprocal Identities to rewrite the fractions
22. 22. And then by using the commutative property of addition…
23. 23. … you’ve successfully proven the identity!
24. 24. One more….
25. 25. Let’s work on the left side of the equation…
26. 26. Multiply each fraction by one to get the LCD
27. 27. Now that the fractions have a common denominator, you can add the numerators
28. 28. Simplify the numerator
29. 29. Use the Pythagorean Identity to rewrite the denominator
30. 30. Multiply the fraction by the constant
31. 31. Use the Reciprocal Identity to rewrite the fraction to equal the expression on the right side of the equation