Question

Verify each equation

Answer 1

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