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  1. \(\cos\left(x+\dfrac\pi2\right)=\cos x\cos\dfrac\pi2-\sin x\sin\dfrac\pi2=-\sin x\)

    2. \(\sin(n\pi+\theta)=\sin(n\pi)\cos\theta+\cos(n\pi)\sin\theta\)

    Since \(\{\sin0,\sin(\pm\pi),\sin(\pm2\pi),\ldots\}\) all amount to 0, the first term disappears. Meanwhile, \(\{\cos0,\cos(\pm\pi),\cos(\pm2\pi),\ldots\}=\{1,-1,1,\ldots\}\) in an alternating pattern, which agrees with the sequence \((-1)^n\). Hence \(\sin(n\pi+\theta)=(-1)^n\sin\theta\) for integers \(n\).

    3. \((\sin x+\cos x)^2=\sin^2x+2\sin x\cos x+\cos^2x=1+2\sin x\cos x=1+\sin2x\)

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