Rút gọn các biểu thức sau:
a) \(\sin x{\cos ^5}x - \cos x{\sin ^5}x\);
b) \(\frac{{\sin 3x\cos 2x + \sin x\cos 6x}}{{\sin 4x}}\);
c) \(\frac{{\cos x - \cos 2x + \cos 3x}}{{\sin x - \sin 2x + \sin 3x}}\);
d) \(\frac{{2\sin \left( {x + y} \right)}}{{\cos \left( {x + y} \right) + \cos \left( {x - y} \right)}} - \tan y\).
Sử dụng kiến thức về các công thức lượng giác để rút gọn:
a) \(\sin 2\alpha = 2\sin \alpha \cos \alpha \), \({\cos ^2}\alpha - {\sin ^2}\alpha = \cos 2\alpha \)
b) \(\sin \alpha \cos \beta = \frac{1}{2}\left[ {\sin \left( {\alpha - \beta } \right) + \sin \left( {\alpha + \beta } \right)} \right]\)
c) \(\cos \alpha + \cos \beta = 2\cos \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2}\), \(\sin \alpha + \sin \beta = 2\sin \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2}\)
d) \(\sin \left( {\alpha + \beta } \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \), \(\cos \alpha + \cos \beta = 2\cos \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2}\)
a) \(\sin x{\cos ^5}x - \cos x{\sin ^5}x \) \( = \sin x\cos x\left( {{{\cos }^4}x - {{\sin }^4}x} \right)\)
\( \) \( = \sin x\cos x\left( {{{\cos }^2}x - {{\sin }^2}x} \right)\left( {{{\cos }^2}x + {{\sin }^2}x} \right) \) \( = \frac{1}{2}\sin 2x\cos 2x \) \( = \frac{1}{4}\sin 4x\)
b) \(\frac{{\sin 3x\cos 2x + \sin x\cos 6x}}{{\sin 4x}} \) \( = \frac{{\frac{1}{2}\left( {\sin 5x + \sin x} \right) + \frac{1}{2}\left( {\sin 7x - \sin 5x} \right)}}{{\sin 4x}}\)
\( \) \( = \frac{{\sin x + \sin 7x}}{{2\sin 4x}} \) \( = \frac{{2\sin 4x\cos 3x}}{{2\sin 4x}} \) \( = \cos 3x\)
c) \(\frac{{\cos x - \cos 2x + \cos 3x}}{{\sin x - \sin 2x + \sin 3x}} \) \( = \frac{{\left( {\cos x + \cos 3x} \right) - \cos 2x}}{{\left( {\sin x + \sin 3x} \right) - \sin 2x}} \) \( = \frac{{2\cos 2x\cos x - \cos 2x}}{{2\sin 2x\cos x - \sin 2x}}\)
\( \) \( = \frac{{\cos 2x\left( {2\cos x - 1} \right)}}{{\sin 2x\left( {2\cos x - 1} \right)}} \) \( = \cot 2x\)
d) \(\frac{{2\sin \left( {x + y} \right)}}{{\cos \left( {x + y} \right) + \cos \left( {x - y} \right)}} - \tan y \) \( = \frac{{2\left( {\sin x\cos y + \cos x\sin y} \right)}}{{2\cos x\cos y}} - \frac{{\sin y}}{{\cos y}}\)
\( \) \( = \frac{{2\sin x\cos y + 2\cos x\sin y - 2\cos x\sin y}}{{2\cos x\cos y}} \) \( = \frac{{2\sin x\cos y}}{{2\cos x\cos y}} \) \( = \tan x\)