\(P=\left[tan\dfrac{17\pi}{4}+tan\left(\dfrac{7\pi}{2}-x\right)\right]^2+\left[cot\dfrac{13\pi}{4}+cot\left(7\pi-x\right)\right]^2\)

\(=\left[tan\dfrac{\pi}{4}+tan\left(-\dfrac{\pi}{2}-x\right)\right]^2+\left[cot\left(-\dfrac{3\pi}{4}\right)+cot\left(-\pi-x\right)\right]^2\)

\(=\left[tan\dfrac{\pi}{4}-cotx\right]^2+\left[tan\dfrac{\pi}{4}-cotx\right]^2\)

\(=2\left(1-cotx\right)^2\)

\(a,\dfrac{1}{tan\alpha+1}+\dfrac{1}{cot\alpha+1}\\ =\dfrac{cot\alpha+1+tan\alpha+1}{\left(tan\alpha+1\right)\left(cot\alpha+1\right)}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha\cdot cot\alpha+tan\alpha+cot\alpha+1}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha+cot\alpha+2}\\ =1\)

\(b,cos\left(\dfrac{\pi}{2}-\alpha\right)-sin\left(\pi+\alpha\right)\\ =sin\alpha+sin\alpha\\ =2sin\alpha\)

\(c,sin\left(\alpha-\dfrac{\pi}{2}\right)+cos\left(-\alpha+6\pi\right)-tan\left(\alpha+\pi\right)cot\left(3\pi-\alpha\right)\\ =-sin\left(\dfrac{\pi}{2}-\alpha\right)+cos\left(\alpha\right)-tan\left(\alpha\right)cot\left(\pi-\alpha\right)\\ =-cos\left(\alpha\right)+cos\left(\alpha\right)+tan\left(\alpha\right)\cdot cot\left(\alpha\right)\\ =1\)

a.

ĐKXĐ: ...

\(\Leftrightarrow tan\left(3x-\frac{\pi}{3}\right)=tan\left(-x\right)\)

\(\Leftrightarrow3x-\frac{\pi}{3}=-x+k\pi\)

\(\Leftrightarrow x=\frac{\pi}{12}+\frac{k\pi}{4}\)

b.

ĐKXĐ: ...

\(\Leftrightarrow cot\left(x-\frac{\pi}{4}\right)=cot\left(-x\right)\)

\(\Leftrightarrow x-\frac{\pi}{4}=-x+k\pi\)

\(\Leftrightarrow x=\frac{\pi}{8}+\frac{k\pi}{2}\)

c.

ĐKXĐ: ...

\(\Leftrightarrow cot\left(2x-\frac{3\pi}{4}\right)=cot\left(\frac{2\pi}{3}-x\right)\)

\(\Leftrightarrow2x-\frac{3\pi}{4}=\frac{2\pi}{3}-x+k\pi\)

\(\Leftrightarrow x=\frac{17\pi}{36}+\frac{k\pi}{3}\)

d.

\(\Leftrightarrow cos\left(2x+\frac{\pi}{3}\right)=cos\left(\frac{3\pi}{4}-x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2x+\frac{\pi}{3}=\frac{3\pi}{4}-x+k2\pi\\2x+\frac{\pi}{3}=x-\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{5\pi}{36}+\frac{k2\pi}{3}\\x=-\frac{13\pi}{12}+k2\pi\end{matrix}\right.\)

\(=cos\left(4\pi+\pi+x\right)+sin\left(4\pi+\frac{\pi}{2}-x\right)-tan\left(\pi+\frac{\pi}{2}+x\right).cot\left(\pi+\frac{\pi}{2}-x\right)\)

\(=cos\left(\pi+x\right)+sin\left(\frac{\pi}{2}-x\right)-tan\left(\frac{\pi}{2}+x\right).cot\left(\frac{\pi}{2}-x\right)\)

\(=-cosx+cosx-\left(-cotx\right).tanx\)

\(=1\)

\(A=2cosx-3cosx-sin\left(3\pi+\frac{\pi}{2}-x\right)+tan\left(\pi+\frac{\pi}{2}-x\right)\)

\(A=-cosx+sin\left(\frac{\pi}{2}-x\right)+tan\left(\frac{\pi}{2}-x\right)\)

\(A=-cosx+cosx+cotx=cotx\)

\(B=2cosx+sin\left(4\pi+\pi-x\right)+sin\left(2\pi-\frac{\pi}{2}+x\right)-sinx\)

\(B=2cosx+sin\left(\pi-x\right)+sin\left(-\frac{\pi}{2}+x\right)-sinx\)

\(B=2cosx+sinx-sin\left(\frac{\pi}{2}-x\right)-sinx\)

\(B=2cosx-cosx=cosx\)

Nhìn đề bài hãi quá :(

a/ \(A=3\sin\left(5.2\pi+\pi-x\right).\sin\left(2\pi+\frac{\pi}{2}-x\right)+2\sin\left(4.2\pi+\pi+x\right)\)

\(A=3\sin\left(\pi-x\right).\sin\left(\frac{\pi}{2}-x\right)+2\sin\left(\pi+x\right)\)

\(A=3\sin x.\cos x-2\sin x=\sin x\left(3\cos x-2\right)\)

b/ \(B=\sin\left(5.2.180^0+180^0+x\right)-\cos\left(90^0-x\right)+\tan\left(90^0+180^0-x\right)+\cot\left(2.180^0-x\right)\)

\(B=\sin\left(180^0+x\right)-\sin x+\tan\left(90^0-x\right)+\cot\left(-x\right)\)

\(B=-\sin x-\sin x+\cot x-\cot x=-2\sin x\)

c/ \(C=-2\sin\left(-(2\pi+\frac{\pi}{2}-x)\right)-3\cos\left(2\pi+\pi-x\right)+5\sin\left(2.2\pi-\left(\frac{\pi}{2}+x\right)\right)+\cot\left(\pi+\frac{\pi}{2}-x\right)\)

\(C=2\sin\left(\frac{\pi}{2}-x\right)-3\cos\left(\pi-x\right)-5\sin\left(\frac{\pi}{2}+x\right)+\cot\left(\frac{\pi}{2}-x\right)\)

\(2\cos x+3\cos x-5\cos x+\tan x=\tan x\)

d/ \(D=\tan\left(-\left(\pi-x\right)\right).\cos\left(-\left(\frac{\pi}{2}-x\right)\right).\left(-\cos x\right)\)

\(D=\tan\left(\pi-x\right).\cos\left(\frac{\pi}{2}-x\right).\cos x\)

\(D=-\tan x.\sin x.\cos x=-\sin^2x\)

e/ \(E=\cos\left(28.2\pi+\pi+\frac{\pi}{2}-x\right)+\sin\left(-\left(58.2\pi+\pi+\frac{\pi}{2}-x\right)\right)+\cos\left(-\left(46.2\pi+\pi+\frac{\pi}{2}-x\right)\right)+\sin\left(35.2\pi+\pi+\frac{\pi}{2}-x\right)\)

\(E=-\cos\left(\frac{\pi}{2}-x\right)+\sin\left(\frac{\pi}{2}-x\right)-\cos\left(\frac{\pi}{2}-x\right)-\sin\left(\frac{\pi}{2}-x\right)\)

\(E=-2\sin x\)

Thôi, stop ở đây, làm nữa chắc tẩu hỏa nhập ma quá :(

Mình thấy hầu hết các bài này đều có chung 1 điểm, và chắc đó cũng là điểm mà bạn thắc mắc: Đó chính là tách các hạng tử ra và biến đổi

Tách cũng đơn giản thôi, cứ gặp sin, cos thì tách sao cho về dạng 2pi+..., gặp tan, cot thì pi.

Còn tách mấy cái phân số như vầy:

Ví dụ \(\frac{7\pi}{2}\) , 7 chia 2 được 3, ta lấy \(\frac{7}{2}-3=\frac{1}{2}\) thì suy ra: \(\frac{7\pi}{2}=3\pi+\frac{\pi}{2}\)

Đó, thế là được :D

\(P=cos\left(-x\right).tanx-cotx.\left(-cotx\right)\)

\(=cosx.tanx+cot^2x=sinx+cot^2x\)

\(=sinx+\frac{1}{1+sin^2x}=-\frac{1}{3}+\frac{1}{1+\frac{1}{9}}=\frac{17}{30}\)

\(A=-sinx-sinx+cot\left(-x\right)=-2sinx-cotx\)