2. 2. Prove the following Identity using two different methods: 2. Prove the following Identity using two different methods:
3. For this question. You can work with the more complicated side. In this case it is the left side. Method # 1: Line of Separation / โGreat Wall of Chinaโ
4. 1} 1} In step one, we realized that we can multiply by reciprocal of the bottom fraction instead of dividing.
5. 2} 2} For step two you can think of tanฮธ as sinฮธ/cosฮธ.
6. 3} In step three we can multiply sinฮธ by sinฮธ/cosฮธ. 3}
7. 4} 4} Here we can multiply sinยฒฮธ/cosฮธ out by 1/cosฮธcosยณฮธ. We can also factor out cosยฒฮธ in the denominator.
8. 5} 5} In this last step we can say that sinยฒฮธ/cosยฒฮธ is tanยฒฮธ and by the Pythagorean identity (1-cosยฒฮธ) is sinยฒฮธ.
11. Method # 2: Line of Separation / โGreat Wall of Chinaโ Working with both sides can also work:
12. 1} Here we can say that tanยฒฮธ is equal to sinยฒฮธ/cosยฒฮธ. ~by saying that, it can be simplified to 1/cosยฒฮธ. 1} [Right Side]
13. 2} 2} Here we recognize cosฮธ-cosยณฮธ as cosฮธ - (cosฮธcosยฒฮธ) or cosฮธ - cosฮธ (1-sinยฒฮธ). Also, tanฮธ as sinฮธ/cosฮธ. [Left Side]
14. 3} 3} In this step we can multiply cosฮธ by (1-sinยฒฮธ).
15. 4} 4} By multiplying by the reciprocal of sinฮธ/cosฮธ we can see that when cosฮธ gets multiplied out, we get cosยฒฮธ-cosยฒฮธ+sinยฒฮธcosฮธ. The two cosยฒฮธโs cancel and youโre left with sinยฒฮธcosฮธ.
16. 5} 5} In this last step we see that sinยฒฮธcosฮธ/sinฮธ leaves us sinฮธcosฮธ. The sinฮธโs reduce leaving us 1/cosฮธ.