Proving-Trigonomddggjhgdsetric-Identities.pptx
- 7. REVIEW 2 2 2 2 2 2 csc 1 cot sec tan 1 1 cos sin Quotient Identities Reciprocal Identities Pythagorean Identities
- 8. (sin 𝑥 )(cot 𝑥 +cos 𝑥 tan 𝑥 )=cos 𝑥 +sin2 𝑥
- 9. x x x x x x 2 sin cos tan cos cot sin x x x x x x cos sin cos sin cos sin Rewrite the terms inside the second parenthesis by using the quotient identities
- 10. x x x x x x 2 sin cos tan cos cot sin x x x x x x cos sin cos sin cos sin Simplify
- 11. x x x x x x 2 sin cos tan cos cot sin x x x x x x sin sin 1 sin sin cos sin To add the fractions inside the parenthesis, you must multiply by one to get common denominators
- 12. x x x x x x 2 sin cos tan cos cot sin Now that you have the common denominators, add the numerators
- 13. x x x x x x 2 sin cos tan cos cot sin x x x x sin sin cos sin 2 Simplify
- 14. x x x x x x 2 sin cos tan cos cot sin x x x x 2 2 sin cos sin cos 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….
- 17. (cos2 𝑥−sin2 𝑥)(cos2 𝑥+sin2 𝑥)=¿ We’ll factor the terms using the difference of two perfect squares technique cos4 𝑥−sin4 𝑥=cos2 𝑥 −sin2 𝑥
- 18. (cos2 𝑥−sin2 𝑥)(1)=¿ Using the Pythagorean Identities the second set of parenthesis can be simplified cos4 𝑥−sin4 𝑥=cos2 𝑥 −sin2 𝑥
- 19. cos2 𝑥−sin2 𝑥=cos2 𝑥−sin2 𝑥 Since the left side of the equation is the same as the right side, you’ve successfully proven the identity! cos4 𝑥−sin4 𝑥=cos2 𝑥 −sin2 𝑥
- 20. On to the next problem….
- 21. tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 22. ¿ cos 𝑥 1− sin 𝑥 (1+sin 𝑥 1+sin 𝑥 ) Multiply by 1 in the form of the conjugate of the denominator. tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 23. ¿ cos 𝑥 (1+sin 𝑥) 1 −sin2 𝑥 Now, let’s distribute in the numerator… tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 24. ¿ cos 𝑥 +cos 𝑥sin 𝑥 cos 2 𝑥 … and simplify the denominator tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 25. ¿ cos 𝑥 cos 2 𝑥 + cos 𝑥 sin 𝑥 cos 2 𝑥 ‘Split’ the fraction and simplify tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 26. ¿ 1 cos 𝑥 + sin 𝑥 cos 𝑥 Use the Quotient and Reciprocal Identities to rewrite the fractions tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 27. ¿ sec 𝑥 +tan 𝑥 And then by using the commutative property of addition… tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 28. tan 𝑥 +sec 𝑥=tan 𝑥+sec 𝑥 … you’ve successfully proven the identity! tan 𝑥+sec 𝑥= cos 𝑥 1−sin 𝑥
- 29. One more….
- 30. 1 1− cos 𝑥 + 1 1+cos 𝑥 =2 csc 2 𝑥
- 31. (1+cos 𝑥 1+cos 𝑥 ) 1 1− cos 𝑥 + 1 1+cos 𝑥 (1−cos 𝑥 1−cos 𝑥 )=¿ Multiply each fraction by one to get the LCD 1 1− cos 𝑥 + 1 1+cos 𝑥 =2 csc 2 𝑥
- 32. 1+cos 𝑥 1− cos 2 𝑥 + 1 −cos 𝑥 1− cos 2 𝑥 =¿ Now that the fractions have a common denominator, you can add the numerators 1 1− cos 𝑥 + 1 1+cos 𝑥 =2 csc 2 𝑥
- 33. 1+cos 𝑥 +1− cos 𝑥 1 −cos 2 𝑥 =¿ Simplify the numerator 1 1− cos 𝑥 + 1 1+cos 𝑥 =2 csc 2 𝑥
- 34. 2 1− cos 2 𝑥 =¿ Use the Pythagorean Identity to rewrite the denominator 1 1− cos 𝑥 + 1 1+cos 𝑥 =2 csc 2 𝑥
- 35. 2 ⋅ 1 sin 2 𝑥 =¿ Multiply the fraction by the constant 1 1− cos 𝑥 + 1 1+cos 𝑥 =2 csc 2 𝑥
- 36. 2 csc2 𝑥=2 csc2 𝑥 Use the Reciprocal Identity to rewrite the fraction to equal the expression on the right side of the equation 1 1− cos 𝑥 + 1 1+cos 𝑥 =2 csc 2 𝑥