Rút gọn các biểu thức sau
a) \(\frac{{\sin ({{45}^0} + \alpha ) - \cos ({{45}^0} + \alpha )}}{{\sin ({{45}^0} + \alpha ) + \cos ({{45}^0} + \alpha )}}\);
b) \(\frac{{\sin 2\alpha + \sin \alpha }}{{1 + \cos 2\alpha + \cos \alpha }}\);
c) \(\frac{{1 + \cos \alpha - \sin \alpha }}{{1 - \cos \alpha - \sin \alpha }}\);
d) \(\frac{{\sin \alpha + \sin 3\alpha + \sin 5\alpha }}{{\cos \alpha + \cos 2\alpha + \cos 5\alpha }}\).
Áp dụng công thức cộng, công thức cơ bản, công thức góc nhân đôi, công thức biên đổi tổng thành tích để biến đổi linh hoạt, rút gọn
\(\begin{array}{l}\cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha .\sin \beta \\\sin (\alpha + \beta ) = \sin \alpha \cos \beta + \cos \alpha .\sin \beta \end{array}\)
\(\frac{{\sin a}}{{\cos a}} = \tan a\); \(\frac{{\cos a}}{{\sin a}} = \cot a\)
\(\cos 2\alpha = 2{\cos ^2}\alpha - 1\);
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\(\sin 2\alpha = 2\sin \alpha \cos \alpha \);
\(\begin{array}{l}\cos a + \cos b = 2\cos \frac{{a + b}}{2}\cos \frac{{a - b}}{2}\\\sin a + \sin b = 2\sin \frac{{a + b}}{2}\cos \frac{{a - b}}{2}\end{array}\)
a) Ta có
\(\begin{array}{l}\frac{{\sin ({{45}^0} + \alpha ) - \cos ({{45}^0} + \alpha )}}{{\sin ({{45}^0} + \alpha ) + \cos ({{45}^0} + \alpha )}}\\ = \frac{{\sin {{45}^0}\cos \alpha + \cos {{45}^0}\sin \alpha - (\cos {{45}^0}\cos \alpha - \sin {{45}^0}\sin \alpha )}}{{\sin {{45}^0}\cos \alpha + \cos {{45}^0}\sin \alpha + (\cos {{45}^0}\cos \alpha - \sin {{45}^0}\sin \alpha )}}\\ = \frac{{\frac{{\sqrt 2 }}{2}\cos \alpha + \frac{{\sqrt 2 }}{2}\sin \alpha - \left( {\frac{{\sqrt 2 }}{2}\cos \alpha - \frac{{\sqrt 2 }}{2}\sin \alpha } \right)}}{{\frac{{\sqrt 2 }}{2}\cos \alpha + \frac{{\sqrt 2 }}{2}\sin \alpha + \left( {\frac{{\sqrt 2 }}{2}\cos \alpha - \frac{{\sqrt 2 }}{2}\sin \alpha } \right)}}\\ = \frac{{\sqrt 2 \sin \alpha }}{{\sqrt 2 \cos \alpha }} = \tan \alpha .\end{array}\)
b) Ta có
\(\begin{array}{l}\frac{{\sin 2\alpha + \sin \alpha }}{{1 + \cos 2\alpha + \cos \alpha }} = \frac{{2\sin \alpha .\cos \alpha + \sin \alpha }}{{1 + 2{{\cos }^2}\alpha - 1 + \cos \alpha }}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\sin \alpha (2\cos \alpha + 1)}}{{2{{\cos }^2}\alpha + \cos \alpha }}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\sin \alpha .(2\cos \alpha + 1)}}{{\cos \alpha .(2\cos \alpha + 1)}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\sin \alpha }}{{\cos \alpha }} = \tan \alpha .\end{array}\)
c) Ta có
\(\begin{array}{l}\frac{{1 + \cos \alpha - \sin \alpha }}{{1 - \cos \alpha - \sin \alpha }}\\ = \frac{{1 + 2{{\cos }^2}\frac{\alpha }{2} - 1 - 2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}}{{1 - \left( {1 - 2{{\sin }^2}\frac{\alpha }{2}} \right) - 2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}}\\ = \frac{{2{{\cos }^2}\frac{\alpha }{2} - 2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}}{{2{{\sin }^2}\frac{\alpha }{2} - 2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}}\\ = \frac{{2\cos \frac{\alpha }{2}.\left( {\cos \frac{\alpha }{2} - \sin \frac{\alpha }{2}} \right)}}{{2\sin \frac{\alpha }{2}.\left( {\sin \frac{\alpha }{2} - \cos \frac{\alpha }{2}} \right)}}\\ = - \frac{{\cos \frac{\alpha }{2}}}{{\sin \frac{\alpha }{2}}} = - \cot \frac{\alpha }{2}.\end{array}\)
d) Ta có:
\(\begin{array}{l}\frac{{\sin \alpha + \sin 3\alpha + \sin 5\alpha }}{{\cos \alpha + \cos 3\alpha + \cos 5\alpha }}\\ = \frac{{\left( {\sin \alpha + \sin 5\alpha } \right) + \sin 3\alpha }}{{\left( {\cos \alpha + \cos 5\alpha } \right) + \cos 3\alpha }}\\ = \frac{{2\sin \frac{{\alpha + 5\alpha }}{2}\cos \frac{{\alpha - 5\alpha }}{2} + \sin 3\alpha }}{{2\cos \frac{{\alpha + 5\alpha }}{2}\cos \frac{{\alpha - 5\alpha + \cos 3\alpha }}{2}}}\\ = \frac{{2\sin 3\alpha .\cos ( - 2\alpha ) + \sin 3\alpha }}{{2\cos 3\alpha \cos ( - 2\alpha ) + \cos 3\alpha }}\\ = \frac{{\sin 3\alpha (2\cos ( - 2\alpha ) + 1)}}{{\cos 3\alpha (2\cos ( - 2\alpha ) + 1)}} = \frac{{\sin 3\alpha }}{{\cos 3\alpha }} = \tan 3\alpha .\end{array}\)