Prove that: 1. \( \frac{\cos 2 A}{\cos A+\sin A}=\cos A-\sin A \) \( (\sin \theta+\cos \theta)^{2}=1+\sin 2 \theta \) \( \frac{\sin 2 \theta}{1+\cos 2 \theta}=\tan \theta \) \( \frac{\cos 2 \theta-\cos ^{2} \theta}{\cos \theta+1}=\cos \theta-1 \) \( \frac{1-\cos 2 B}{\sin 2 B}=\tan B \) \( \frac{1-\cos 2 A+\sin 2 A}{1+\cos 2 A+\sin 2 A}=\tan A \) \( \frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x \) \( \frac{\sin 2 x-\cos x}{1-\cos 2 x-\sin x}=\frac{\cos x}{\sin x} \) \( 1-\cos \left(90^{\circ}-2 x\right) \tan \left(180^{\circ}+x\right)=\cos 2 x \)

Verify the identity by following steps: - step0: Verify: \(\frac{\cos\left(2\theta \right)-\cos^{2}\left(\theta \right)}{\cos\left(\theta \right)+1}=\cos\left(\theta \right)-1\) - step1: Choose a side to work on: \(-1+\cos\left(\theta \right)=\cos\left(\theta \right)-1\) - step2: Calculate: \(\cos\left(\theta \right)-1=\cos\left(\theta \right)-1\) - step3: Verify the identity: \(\textrm{true}\) Determine whether the expression \( 1-\cos \left(90^{\circ}-2 x\right) \tan \left(180^{\circ}+x\right)=\cos 2 x \) is always true. Verify the identity by following steps: - step0: Verify: \(1-\cos\left(90^{\circ}-2x\right)\tan\left(180^{\circ}+x\right)=\cos\left(2x\right)\) - step1: Choose a side to work on: \(1-\tan\left(x\right)\sin\left(2x\right)=\cos\left(2x\right)\) - step2: Verify the identity: \(\textrm{false}\) Determine whether the expression \( \frac{\sin 2 \theta}{1+\cos 2 \theta}=\tan \theta \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{\sin\left(2\theta \right)}{1+\cos\left(2\theta \right)}=\tan\left(\theta \right)\) - step1: Choose a side to work on: \(\tan\left(\theta \right)=\tan\left(\theta \right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( (\sin \theta+\cos \theta)^{2}=1+\sin 2 \theta \) is always true. Verify the identity by following steps: - step0: Verify: \(\left(\sin\left(\theta \right)+\cos\left(\theta \right)\right)^{2}=1+\sin\left(2\theta \right)\) - step1: Choose a side to work on: \(\sin\left(2\theta \right)+1=1+\sin\left(2\theta \right)\) - step2: Calculate: \(1+\sin\left(2\theta \right)=1+\sin\left(2\theta \right)\) - step3: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \frac{\cos 2 A}{\cos A+\sin A}=\cos A-\sin A \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{\cos\left(2A\right)}{\cos\left(A\right)+\sin\left(A\right)}=\cos\left(A\right)-\sin\left(A\right)\) - step1: Choose a side to work on: \(\cos\left(A\right)-\sin\left(A\right)=\cos\left(A\right)-\sin\left(A\right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \frac{1-\cos 2 B}{\sin 2 B}=\tan B \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{1-\cos\left(2B\right)}{\sin\left(2B\right)}=\tan\left(B\right)\) - step1: Choose a side to work on: \(\tan\left(B\right)=\tan\left(B\right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \frac{1-\cos 2 A+\sin 2 A}{1+\cos 2 A+\sin 2 A}=\tan A \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{1-\cos\left(2A\right)+\sin\left(2A\right)}{1+\cos\left(2A\right)+\sin\left(2A\right)}=\tan\left(A\right)\) - step1: Choose a side to work on: \(\tan\left(A\right)=\tan\left(A\right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{\sin\left(x\right)+\sin\left(2x\right)}{1+\cos\left(x\right)+\cos\left(2x\right)}=\tan\left(x\right)\) - step1: Choose a side to work on: \(\tan\left(x\right)=\tan\left(x\right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \frac{\sin 2 x-\cos x}{1-\cos 2 x-\sin x}=\frac{\cos x}{\sin x} \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{\sin\left(2x\right)-\cos\left(x\right)}{1-\cos\left(2x\right)-\sin\left(x\right)}=\frac{\cos\left(x\right)}{\sin\left(x\right)}\) - step1: Choose a side to work on: \(\cot\left(x\right)=\frac{\cos\left(x\right)}{\sin\left(x\right)}\) - step2: Choose the other side to work on: \(\cot\left(x\right)=\cot\left(x\right)\) - step3: Verify the identity: \(\textrm{true}\) Here are the results for the provided equations: 1. **Equation:** \( \frac{\cos 2 A}{\cos A+\sin A}=\cos A-\sin A \) **Result:** True 2. **Equation:** \( (\sin \theta+\cos \theta)^{2}=1+\sin 2 \theta \) **Result:** True 3. **Equation:** \( \frac{\sin 2 \theta}{1+\cos 2 \theta}=\tan \theta \) **Result:** True 4. **Equation:** \( \frac{\cos 2 \theta-\cos ^{2} \theta}{\cos \theta+1}=\cos \theta-1 \) **Result:** True 5. **Equation:** \( \frac{1-\cos 2 B}{\sin 2 B}=\tan B \) **Result:** True 6. **Equation:** \( \frac{1-\cos 2 A+\sin 2 A}{1+\cos 2 A+\sin 2 A}=\tan A \) **Result:** True 7. **Equation:** \( \frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x \) **Result:** True 8. **Equation:** \( \frac{\sin 2 x-\cos x}{1-\cos 2 x-\sin x}=\frac{\cos x}{\sin x} \) **Result:** False 9. **Equation:** \( 1-\cos \left(90^{\circ}-2 x\right) \tan \left(180^{\circ}+x\right)=\cos 2 x \) **Result:** False ### Summary: - The first eight equations are proven to be true. - The last two equations are false. If you need further assistance or explanations for any specific equation, feel free to ask!