42-58 Prove the identity. 42. \( \cos \left(\frac{\pi}{2}-x\right)=\sin x \) 43. \( \sin \left(\frac{\pi}{2}+x\right)=\cos x \) 44. \( \sin (\pi-x)=\sin x \) 45. \( \sin \theta \cot \theta=\cos \theta \) 46. \( (\sin x+\cos x)^{2}=1+\sin 2 x \) 47. \( \sec y-\cos y=\tan y \sin y \) 48. \( \tan 2 \alpha-\sin ^{2} \alpha=\tan ^{2} \alpha \sin ^{2} \alpha \) 49. \( \cot ^{2} \theta+\sec ^{2} \theta=\tan ^{2} \theta+\csc ^{2} \theta \) 50. \( 2 \csc 2 t=\sec ^{2} \csc ^{2} \)

Verify the identity by following steps: - step0: Verify: \(\sin\left(\frac{\pi }{2}+x\right)=\cos\left(x\right)\) - step1: Choose a side to work on: \(\cos\left(x\right)=\cos\left(x\right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \sin \theta \cot \theta=\cos \theta \) is always true. Verify the identity by following steps: - step0: Verify: \(\sin\left(\theta \right)\cot\left(\theta \right)=\cos\left(\theta \right)\) - step1: Choose a side to work on: \(\cos\left(\theta \right)=\cos\left(\theta \right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \sin (\pi-x)=\sin x \) is always true. Verify the identity by following steps: - step0: Verify: \(\sin\left(\pi -x\right)=\sin\left(x\right)\) - step1: Choose a side to work on: \(\sin\left(x\right)=\sin\left(x\right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \sec y-\cos y=\tan y \sin y \) is always true. Verify the identity by following steps: - step0: Verify: \(\sec\left(y\right)-\cos\left(y\right)=\tan\left(y\right)\sin\left(y\right)\) - step1: Choose a side to work on: \(\frac{\sin^{2}\left(y\right)}{\cos\left(y\right)}=\tan\left(y\right)\sin\left(y\right)\) - step2: Choose the other side to work on: \(\frac{\sin^{2}\left(y\right)}{\cos\left(y\right)}=\frac{\sin^{2}\left(y\right)}{\cos\left(y\right)}\) - step3: Verify the identity: \(\textrm{true}\) Determine whether the expression \( (\sin x+\cos x)^{2}=1+\sin 2 x \) is always true. Verify the identity by following steps: - step0: Verify: \(\left(\sin\left(x\right)+\cos\left(x\right)\right)^{2}=1+\sin\left(2x\right)\) - step1: Choose a side to work on: \(\sin\left(2x\right)+1=1+\sin\left(2x\right)\) - step2: Calculate: \(1+\sin\left(2x\right)=1+\sin\left(2x\right)\) - step3: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \cos \left(\frac{\pi}{2}-x\right)=\sin x \) is always true. Verify the identity by following steps: - step0: Verify: \(\cos\left(\frac{\pi }{2}-x\right)=\sin\left(x\right)\) - step1: Choose a side to work on: \(\sin\left(x\right)=\sin\left(x\right)\) - step2: Verify the identity: \(\textrm{true}\) Determine whether the expression \( \cot ^{2} \theta+\sec ^{2} \theta=\tan ^{2} \theta+\csc ^{2} \theta \) is always true. Verify the identity by following steps: - step0: Verify: \(\cot^{2}\left(\theta \right)+\sec^{2}\left(\theta \right)=\tan^{2}\left(\theta \right)+\csc^{2}\left(\theta \right)\) - step1: Choose a side to work on: \(\frac{\cos^{2}\left(\theta \right)+\sec^{2}\left(\theta \right)\sin^{2}\left(\theta \right)}{\sin^{2}\left(\theta \right)}=\tan^{2}\left(\theta \right)+\csc^{2}\left(\theta \right)\) - step2: Choose the other side to work on: \(\frac{\cos^{2}\left(\theta \right)+\sec^{2}\left(\theta \right)\sin^{2}\left(\theta \right)}{\sin^{2}\left(\theta \right)}=\frac{\sin^{2}\left(\theta \right)+\csc^{2}\left(\theta \right)\cos^{2}\left(\theta \right)}{\cos^{2}\left(\theta \right)}\) - step3: Verify the identity: \(\textrm{false}\) Here are the results for the identities you provided: 1. **Identity**: \( \cos \left(\frac{\pi}{2}-x\right)=\sin x \) **Result**: True 2. **Identity**: \( \sin \left(\frac{\pi}{2}+x\right)=\cos x \) **Result**: True 3. **Identity**: \( \sin (\pi-x)=\sin x \) **Result**: True 4. **Identity**: \( \sin \theta \cot \theta=\cos \theta \) **Result**: True 5. **Identity**: \( (\sin x+\cos x)^{2}=1+\sin 2 x \) **Result**: True 6. **Identity**: \( \sec y-\cos y=\tan y \sin y \) **Result**: True 7. **Identity**: \( \tan 2 \alpha-\sin ^{2} \alpha=\tan ^{2} \alpha \sin ^{2} \alpha \) **Result**: Calculation timeout. This identity may be too complex to evaluate directly. 8. **Identity**: \( \cot ^{2} \theta+\sec ^{2} \theta=\tan ^{2} \theta+\csc ^{2} \theta \) **Result**: False 9. **Identity**: \( 2 \csc 2 t=\sec ^{2} \csc ^{2} \) **Result**: Unsupported calculation. If you need further assistance with the identities that timed out or were unsupported, please let me know!