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

Ask by O'Quinn Carter. in South Africa
Mar 11,2025

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Tutor-Verified Answer

Answer

The first eight equations are true, but the last two are false.

Solution

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!

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The Deep Dive

Trigonometric identities have a rich historical background, dating back to ancient civilizations. In India, mathematicians like Aryabhata developed sine and cosine ratios by studying the movement of celestial bodies around the 5th century. These early discoveries laid the groundwork for modern trigonometry, allowing us to solve complex equations, navigate by stars, and even design structures. In real-world applications, trigonometric identities play a vital role beyond mathematics. Engineers use them in physics to calculate forces and motion. Architects rely on these equations for designing stable structures and optimizing light exposure. Even in computer graphics, trigonometric functions help create realistic animations and visual effects, bringing games and movies to life in ways that enthrall audiences!

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