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. 1} 2} 3} 4} 5} QED 1} In step one, we realized that we can multiply by the bottom fraction instead of divide. 2} For step two you can think of tanθ as sinθ/cosθ. 3} In step three we can multiply sinθ by sinθ/cosθ. 4} Here we can multiply sin²θ/cosθ out by 1/cosθcos³θ. 5} In this last step we can say that sin²θ/cos²θ is tan²θ and by the Pythagorean identity (1-cos²θ) is sin²θ. Method # 1:
4. Method # 2: QED Working with both sides can also work: 1} 2} 3} 4} 5} 1} Here we can say that tan²θ is equal to sin²θ/cos²θ. ~by saying that, it can be simplified to 1/cos²θ. 2} Here we can recognize cosθ-cos³θ as cosθ - (cosθcos²θ) or cosθ - cosθ (1-sin²θ). Also, tanθ as sinθ/cosθ. 3} In this step we can multiply cosθ by (1-sin²θ). 4} By multiplying by the reciprocal of sinθ/cosθ we can see that when cosθ gets multiplied out, we get cos²θ-cos²θ+sin²θcosθ. The two cos²θ’s cancel and you’re left with sin²θcosθ. 5} In this last step we see that sin²θcosθ/sinθ leaves us sinθcosθ. The sinθ’s reduce leaving us 1/cosθ.
