3.2:
a: \(A=cos\left(\alpha-\frac{\pi}{2}\right)+\sin\left(\alpha-\pi\right)\)
\(=cos\alpha\cdot cos\left(\frac{\pi}{2}\right)+\sin\left(\alpha\right)\cdot\sin\left(\frac{\pi}{2}\right)+\sin\alpha\cdot cos\left(\pi\right)-\sin\pi\cdot cos\alpha\)
\(=\sin\alpha-\sin\alpha=0\)
b: \(B=cos\left(\pi-\alpha\right)+\sin\left(\alpha+\frac{\pi}{2}\right)\)
\(=cos\pi\cdot cos\alpha+\sin\pi\cdot\sin\alpha+\sin\alpha\cdot cos\left(\frac{\pi}{2}\right)+\sin\left(\frac{\pi}{2}\right)\cdot cos\alpha\)
\(=-cos\alpha+cos\alpha=0\)
c: \(C=cos\left(\frac{\pi}{2}-\alpha\right)+\sin\left(\frac{\pi}{2}-\alpha\right)-cos\left(\frac{\pi}{2}+\alpha\right)\)
\(=\sin\alpha+cos\alpha-\left(cos\left(\frac{\pi}{2}\right)\cdot cos\alpha-\sin\left(\frac{\pi}{2}\right)\cdot\sin\alpha\right)\)
\(=\sin\alpha+cos\alpha+\sin\alpha=2\cdot\sin\alpha+cos\alpha\)
d: \(D=\tan\left(\alpha-\pi\right)\cdot cos\left(\alpha-\pi\right)\cdot cos\left(\alpha-\frac{\pi}{2}\right)\)
\(=\sin\left(\alpha-\pi\right)\cdot cos\left(\frac{\pi}{2}-\alpha\right)\)
\(=\left(\sin\alpha\cdot cos\pi-\sin\pi\cdot cos\alpha\right)\cdot\sin\alpha=\left(-\sin\alpha\right)\cdot\sin\alpha=-\sin^2\alpha\)
e: \(E=\cot\left(\frac{\pi}{2}-\alpha\right)\cdot\sin\left(\alpha-\frac{\pi}{2}\right)\)
\(=\tan\alpha\cdot\left\lbrack-\sin\left(\frac{\pi}{2}-\alpha\right)\right\rbrack=\tan\alpha\cdot\left(-cos\alpha\right)=-\sin\alpha\)
f: \(F=\sin\left(\pi+x\right)-cos\left(\frac{\pi}{2}-x\right)+\tan\left(\frac32\pi-x\right)+\cot\left(2\pi-x\right)\)
\(=\sin\pi\cdot cosx+cos\pi\cdot\sin x-\sin x+\tan\left(\pi+\frac{\pi}{2}-x\right)+\cot\left(-x\right)\)
\(=-\sin x-\sin x+\tan\left(\frac{\pi}{2}-x\right)-\cot x\)
\(=-2\cdot\sin x+\cot x-\cot x=-2\cdot\sin x\)
g: \(G=\cot\left(\alpha-2\pi\right)\cdot cos\left(\alpha-\frac32\pi\right)+cos\left(\alpha+2\pi\right)-3\cdot\sin\left(\alpha-\pi\right)\)
\(=\cot\alpha\cdot\left\lbrack cos\alpha\cdot cos\left(\frac32\pi\right)+\sin\alpha\cdot\sin\left(\frac32\pi\right)\right\rbrack+cos\alpha-3\cdot\left(\sin\alpha\cdot cos\pi-cos\alpha\cdot\sin\pi\right)\)
\(=\cot\alpha\cdot\left(-\sin\alpha\right)+cos\alpha-3\cdot\left(-\sin\alpha\right)=-cos\alpha+cos\alpha+3\cdot\sin\alpha=3\cdot\sin\alpha\)
h: \(H=cos\left(\frac32\pi-\alpha\right)-\sin\left(\frac32\pi-\alpha\right)+cos\left(\alpha-\frac72\pi\right)-\sin\left(a-\frac72\pi\right)\)
\(=cos\left(2\pi-\frac{\pi}{2}-\alpha\right)-\sin\left(2\pi-\frac{\pi}{2}-\alpha\right)+cos\left(a-\frac32\pi-2\pi\right)-\sin\left(\alpha-\frac32\pi-2\pi\right)\)
\(=cos\left(-\frac{\pi}{2}-\alpha\right)-\sin\left(-\frac{\pi}{2}-\alpha\right)+cos\left(\alpha-\frac32\pi\right)-\sin\left(a-\frac32\pi\right)\)
\(=cos\left(\frac{\pi}{2}+\alpha\right)+\sin\left(\frac{\pi}{2}+\alpha\right)+cos\alpha\cdot cos\left(\frac32\pi\right)+\sin\alpha\cdot\sin\left(\frac32\pi\right)-\sin\alpha\cdot cos\left(\frac32\pi\right)+cos\alpha\cdot\sin\left(\frac32\pi\right)\)
\(=cos\left(\frac{\pi}{2}+\alpha\right)+\sin\left(\frac{\pi}{2}+\alpha\right)-\sin\alpha-cos\alpha\)
\(=cos\left(\frac{\pi}{2}\right)\cdot cos\alpha-\sin\left(\frac{\pi}{2}\right)\cdot\sin\alpha+\sin\left(\frac{\pi}{2}\right)\cdot cos\alpha+cos\left(\frac{\pi}{2}\right)\cdot\sin\alpha-\sin\alpha-cos\alpha\)
\(=-\sin\alpha+cos\alpha-\sin\alpha-cos\alpha=-2\cdot\sin\alpha\)
Bài 3.3:
a: \(E=\sin\left(x+\pi\right)+\sin\left(x+2\pi\right)+\cdots+\sin\left(x+k2\pi\right)\)
\(=-\sin x+\sin x-\sin x+\sin x-\cdots+\sin x\)
=0
b: \(F=2\cdot cosx-3\cdot cos\left(\pi-x\right)+5\cdot\sin\left(\frac72\pi-x\right)+\cot\left(\frac32\pi-x\right)\)
\(=2\cdot cosx-3\cdot\left\lbrack cos\pi\cdot cosx+\sin\pi\cdot\sin x\right\rbrack+5\cdot\sin\left(2\pi+\frac32\pi-x\right)+\cot\left(\frac32\pi-x\right)\)
\(=2\cdot cosx-3\cdot\left\lbrack-cosx\right\rbrack+5\cdot\sin\left(\frac32\pi-x\right)+\cot\left(\pi+\frac{\pi}{2}-x\right)\)
\(=5\cdot cosx+3\cdot\left\lbrack\sin\left(\frac32\pi\right)\cdot cosx-cos\left(\frac32\pi\right)\cdot\sin x\right\rbrack+\cot\left(\frac{\pi}{2}-x\right)\)
\(=5\cdot cosx+3\cdot\left\lbrack-cosx\right\rbrack+\tan x=2\cdot cosx+\tan x\)
