Biến đổi thành tích :
a) \(1+\cos\left(\dfrac{\pi}{2}+3\alpha\right)-\sin\left(\dfrac{3}{2}\pi-3\alpha\right)+\cot\left(\dfrac{5}{2}\pi+3\alpha\right)\)
b) \(\dfrac{\cos7\alpha-\cos8\alpha-\cos9\alpha+\cos10\alpha}{\sin7\alpha-\sin8\alpha-\sin9\alpha+\sin10\alpha}\)
c) \(-\cos5a.\cos4a-\cos4a\cos3a+2\cos^22a\cos a\)
Chứng minh các hệ thức sau :
a) \(\sin\alpha+\sin\left(\alpha+\dfrac{14}{3}\pi\right)+\sin\left(\alpha-\dfrac{8}{3}\pi\right)=0\)
b) \(\dfrac{\sin4a}{1+\cos4a}.\dfrac{\cos2a}{1+\cos2a}=\cot\left(\dfrac{3}{2}\pi-a\right)\)
c) \(\left(\cos a-\cos b\right)^2-\left(\sin a-\sin b\right)^2=-4\sin^2\dfrac{a-b}{2}\cos\left(a+b\right)\)
d) \(\sin^2\left(45^0+\alpha\right)-\sin^2\left(30^0-\alpha\right)-\sin15^0\cos\left(15^0+2\alpha\right)=\sin2\alpha\)
Cho góc \(\alpha\) thỏa mãn \(sin\alpha=\dfrac{4}{5}\) . Tính \(P=cos4\alpha\)
Giải các phương trình :
a) \(\cos\left(22^0-t\right)\cos\left(82^0-t\right)+\cos\left(112^0-t\right)\cos\left(172^0-t\right)=\dfrac{1}{2}\left(\sin t+\cos t\right)\)
b) \(\sin^2\left(t+45^0\right)-\sin^2\left(t-30^0\right)-\sin15^0\cos\left(2t+15^0\right)=\dfrac{1}{2}\sin6t\)
c) \(\sin^82x+\cos^82x=\dfrac{41}{128}\)
d) \(\sqrt{4\cos^2+1}+\sqrt{4\sin^2x+3}=4\)
e) \(\tan\left(\pi\cot t\right)=\cot\left(\pi\sin t\right)\)
Giải các phương trình :
a) \(\cos^2x+\cos^22x-\cos^23x-\cos^24x=0\)
b) \(\cos4x\cos\left(\pi+2x\right)-\sin2x\cos\left(\dfrac{\pi}{2}-4x\right)=\dfrac{\sqrt{2}}{2}\sin4x\)
c) \(\tan\left(120^0+3x\right)-\tan\left(140^0-x\right)=2\sin\left(80^0+2x\right)\)
d) \(\tan^2\dfrac{x}{2}+\sin^2\dfrac{x}{2}\tan\dfrac{x}{2}+\cos^2\dfrac{x}{2}+\cot^2\dfrac{x}{2}+\sin x=4\)
e) \(\dfrac{\sin2t+2\cos^2t-1}{\cot t-\cot3t+\sin3t-\sin t}=\cos t\)
Giải các pt sau :
a, \(sin\dfrac{x}{2}\cdot sinx-cos\dfrac{x}{2}\cdot sin^2x+1-2cos^2\cdot\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)=0\)
b, \(tanx-3cotx=4\cdot\left(sinx+\sqrt{3}\cdot cosx\right)\)
Tìm nghiệm \(x\in\left(0;10\pi\right)\) của phương trình
\(\dfrac{\sqrt{3}}{cos^2x}-tanx-2\sqrt{3}=sinx\left(1+tanx.tan\dfrac{x}{2}\right).\)
Tính đạo hàm của các hàm số sau :
a) \(y=\dfrac{1+x-x^2}{1-x+x^2}\)
b) \(y=\dfrac{\left(2-x^2\right)\left(3-x^3\right)}{\left(1-x\right)^2}\)
c) \(y=\cos2x-2\sin x\)
d) \(y=\dfrac{\cos x}{2\sin^2x}\)
e) \(y=\cos^2\dfrac{x}{3}\tan\dfrac{x}{2}\)
f) \(y=\sqrt{\sin\left(2x-\dfrac{\pi}{6}\right)}\)
g) \(y=\cos\dfrac{x}{x+1}\)
h) \(y=\dfrac{x^2-1}{\sin3x}\)
i) \(y=3\sin^2x\cos x+\cos^2x\)
k) \(y=\sqrt{7-4x}\cot3x\)
Giải: \(2\sqrt{2}cos^3\left(x-\dfrac{\pi}{4}\right)-3cosx-sinx=0\)
Giải các phương trình :
a) \(\sin2x=\cos^4\dfrac{x}{2}-\sin^4\dfrac{x}{2}\)
b) \(3\sin5x-2\cos5x=3\)
c) \(\cos\left(\dfrac{\pi}{2}+5x\right)+\sin x=2\cos3x\)
d) \(\sin2z+\cos2z=\sqrt{2}\sin3z\)
