8. Prove that: \( \frac{1-\cos ^{2} \theta}{\cos ^{2} \theta+2 \cos \theta+1}=\frac{1-\cos \theta}{1+\cos \theta} \) 10. Prove that: \( \frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=\frac{4 \tan x}{\cos x} \) 11. Prove that: \( \frac{1+2 \sin \theta \cdot \cos \theta}{\sin \theta+\cos \theta}=\sin \theta+\cos \theta \) (consider number 6 )

Verify the identity by following steps: - step0: Verify: \(\frac{1-\cos^{2}\left(\theta \right)}{\cos^{2}\left(\theta \right)+2\cos\left(\theta \right)+1}=\frac{1-\cos\left(\theta \right)}{1+\cos\left(\theta \right)}\) - step1: Choose a side to work on: \(\frac{1}{\cos\left(\theta \right)+1}-\frac{\cos\left(\theta \right)}{\cos\left(\theta \right)+1}=\frac{1-\cos\left(\theta \right)}{1+\cos\left(\theta \right)}\) - step2: Choose the other side to work on: \(\frac{1}{\cos\left(\theta \right)+1}-\frac{\cos\left(\theta \right)}{\cos\left(\theta \right)+1}=\frac{1}{1+\cos\left(\theta \right)}-\frac{\cos\left(\theta \right)}{1+\cos\left(\theta \right)}\) - step3: Calculate: \(\frac{1}{1+\cos\left(\theta \right)}-\frac{\cos\left(\theta \right)}{1+\cos\left(\theta \right)}=\frac{1}{1+\cos\left(\theta \right)}-\frac{\cos\left(\theta \right)}{1+\cos\left(\theta \right)}\) - step4: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \frac{1+2 \sin \theta \cdot \cos \theta}{\sin \theta+\cos \theta}=\sin \theta+\cos \theta \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{1+2\sin\left(\theta \right)\cos\left(\theta \right)}{\sin\left(\theta \right)+\cos\left(\theta \right)}=\sin\left(\theta \right)+\cos\left(\theta \right)\) - step1: Choose a side to work on: \(\frac{1}{\sin\left(\theta \right)+\cos\left(\theta \right)}+\frac{\sin\left(2\theta \right)}{\sin\left(\theta \right)+\cos\left(\theta \right)}=\sin\left(\theta \right)+\cos\left(\theta \right)\) - step2: Verify the identity: \(\textrm{false}\) Determine whether the expression \( \frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=\frac{4 \tan x}{\cos x} \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}-\frac{1-\sin\left(x\right)}{1+\sin\left(x\right)}=\frac{4\tan\left(x\right)}{\cos\left(x\right)}\) - step1: Choose a side to work on: \(\frac{4\sin\left(x\right)}{\cos^{2}\left(x\right)}=\frac{4\tan\left(x\right)}{\cos\left(x\right)}\) - step2: Choose the other side to work on: \(\frac{4\sin\left(x\right)}{\cos^{2}\left(x\right)}=\frac{4\sin\left(x\right)}{\cos^{2}\left(x\right)}\) - step3: Verify the identity: \(\textrm{true}\) Here are the results for the three proofs: 1. **First Proof:** \[ \frac{1-\cos^{2} \theta}{\cos^{2} \theta+2 \cos \theta+1} = \frac{1-\cos \theta}{1+\cos \theta} \] - This statement is **true**. 2. **Second Proof:** \[ \frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x} = \frac{4 \tan x}{\cos x} \] - This statement is **true**. 3. **Third Proof:** \[ \frac{1+2 \sin \theta \cdot \cos \theta}{\sin \theta+\cos \theta} = \sin \theta+\cos \theta \] - This statement is **false**. If you need further clarification or additional proofs, feel free to ask!