d. \( \left(1-\cos ^{2} A\right)\left(1+\tan ^{2} A\right)=\tan ^{2} A \) e. \( \cos ^{4} \theta+\sin ^{4} \theta=1-2 \sin ^{2} \theta \cos ^{2} \theta \) f. \( \sec ^{4} A-\sec ^{2} A=\tan ^{4} A+\tan ^{2} A \) g. \( \cos A \operatorname{cosec} A \sqrt{\sec ^{2} \theta-1}=1 \) h. \( \frac{\cos A}{1-\sin A}+\frac{\cos A}{1+\sin A}=2 \sec A \) i. \( \frac{1}{1-\cos A}+\frac{1}{1+\cos A}=2 \operatorname{cosec}^{2} A \) j. \( \frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}=\frac{1+\tan \theta}{1-\tan \theta} \) k. \( \frac{1-\tan \beta}{1+\tan \beta}=\frac{\cot \beta-1}{\cot \beta+1} \) 1. \( \frac{\tan \alpha+\tan \beta}{\cot \alpha+\cot \beta}=\tan \alpha \tan \beta \) m. \( \frac{1-\sin \alpha}{\cos \alpha}=1+2 \tan ^{2} \alpha \)

Verify the identity by following steps: - step0: Verify: \(\cos^{4}\left(\theta \right)+\sin^{4}\left(\theta \right)=1-2\sin^{2}\left(\theta \right)\cos^{2}\left(\theta \right)\) - step1: Choose a side to work on: \(1-\frac{\sin^{2}\left(2\theta \right)}{2}=1-2\sin^{2}\left(\theta \right)\cos^{2}\left(\theta \right)\) - step2: Choose the other side to work on: \(1-\frac{\sin^{2}\left(2\theta \right)}{2}=1-\frac{\sin^{2}\left(2\theta \right)}{2}\) - step3: Verify the identity: \(\textrm{true}\) Determine whether the expression \( (\cos A)/(1-\sin A)+(\cos A)/(1+\sin A)=2 \sec A \) is always true. Verify the identity by following steps: - step0: Verify: \(\frac{\left(\cos\left(A\right)\right)}{\left(1-\sin\left(A\right)\right)}+\frac{\left(\cos\left(A\right)\right)}{\left(1+\sin\left(A\right)\right)}=2\sec\left(A\right)\) - step1: Choose a side to work on: \(\frac{2}{\cos\left(A\right)}=2\sec\left(A\right)\) - step2: Choose the other side to work on: \(\frac{2}{\cos\left(A\right)}=\frac{2}{\cos\left(A\right)}\) - step3: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \sec^4 A-\sec^2 A=\tan^4 A+\tan^2 A \) is always true. Verify the identity by following steps: - step0: Verify: \(\sec^{4}\left(A\right)-\sec^{2}\left(A\right)=\tan^{4}\left(A\right)+\tan^{2}\left(A\right)\) - step1: Choose a side to work on: \(-\frac{\sin^{2}\left(A\right)}{\left(-\cos^{2}\left(A\right)\right)^{2}}=\tan^{4}\left(A\right)+\tan^{2}\left(A\right)\) - step2: Choose the other side to work on: \(-\frac{\sin^{2}\left(A\right)}{\left(-\cos^{2}\left(A\right)\right)^{2}}=-\frac{\sin^{2}\left(A\right)}{\left(-\cos^{2}\left(A\right)\right)^{2}}\) - step3: Verify the identity: \(\textrm{true}\) Determine whether the expression \( (1-\cos^2 A)(1+\tan^2 A)=\tan^2 A \) is always true. Verify the identity by following steps: - step0: Verify: \(\left(1-\cos^{2}\left(A\right)\right)\left(1+\tan^{2}\left(A\right)\right)=\tan^{2}\left(A\right)\) - step1: Choose a side to work on: \(\tan^{2}\left(A\right)=\tan^{2}\left(A\right)\) - step2: Verify the identity: \(\textrm{true}\) Here are the results for the provided equations: 1. **Equation d:** \[ (1 - \cos^2 A)(1 + \tan^2 A) = \tan^2 A \] - **Result:** True 2. **Equation e:** \[ \cos^4 \theta + \sin^4 \theta = 1 - 2 \sin^2 \theta \cos^2 \theta \] - **Result:** True 3. **Equation f:** \[ \sec^4 A - \sec^2 A = \tan^4 A + \tan^2 A \] - **Result:** True 4. **Equation g:** \[ \cos A \operatorname{cosec} A \sqrt{\sec^2 \theta - 1} = 1 \] - **Result:** Service not available for this calculation. 5. **Equation h:** \[ \frac{\cos A}{1 - \sin A} + \frac{\cos A}{1 + \sin A} = 2 \sec A \] - **Result:** True 6. **Equation i:** \[ \frac{1}{1 - \cos A} + \frac{1}{1 + \cos A} = 2 \operatorname{cosec}^2 A \] - **Result:** Calculation timeout. The calculation may be too complex. 7. **Equation j:** \[ \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} = \frac{1 + \tan \theta}{1 - \tan \theta} \] - **Result:** Service not available for this calculation. 8. **Equation k:** \[ \frac{1 - \tan \beta}{1 + \tan \beta} = \frac{\cot \beta - 1}{\cot \beta + 1} \] - **Result:** Service not available for this calculation. 9. **Equation l:** \[ \frac{\tan \alpha + \tan \beta}{\cot \alpha + \cot \beta} = \tan \alpha \tan \beta \] - **Result:** Service not available for this calculation. 10. **Equation m:** \[ \frac{1 - \sin \alpha}{\cos \alpha} = 1 + 2 \tan^2 \alpha \] - **Result:** Service not available for this calculation. ### Summary: - Equations d, e, f, and h are confirmed to be true. - Equations g, i, j, k, l, and m could not be evaluated due to service unavailability or timeout issues. If you need further assistance or specific evaluations, please let me know!