Chứng minh rằng 1 /cos 4 x + tan 4 x = (cos 2 x + sin 2 x)/( cos 2 x − sin 2 x) .
Giải thích
Lời giải:
\(\frac{1}{{\cos 4{\rm{x}}}} + \tan 4{\rm{x}}\)
\( = \frac{1}{{\cos 4{\rm{x}}}} + \frac{{\tan 4{\rm{x}} \cdot \cos 4{\rm{x}}}}{{\cos 4{\rm{x}}}}\)
\( = \frac{1}{{\cos 4{\rm{x}}}} + \frac{{\frac{{\sin 4{\rm{x}}}}{{\cos 4{\rm{x}}}} \cdot \cos 4{\rm{x}}}}{{\cos 4{\rm{x}}}}\)
\( = \frac{{1 + \sin 4{\rm{x}}}}{{\cos 4{\rm{x}}}}\)
\( = \frac{{{{\sin }^2}2{\rm{x}} + {{\cos }^2}2{\rm{x}} + 2\sin 2{\rm{x}} \cdot \cos 2{\rm{x}}}}{{{{\sin }^2}2{\rm{x}} - {{\cos }^2}2{\rm{x}}}}\)
\( = \frac{{{{\left( {\sin 2{\rm{x}} + \cos 2{\rm{x}}} \right)}^2}}}{{\left( {\sin 2{\rm{x}} + \cos 2{\rm{x}}} \right)\left( {\sin 2{\rm{x}} - \cos 2{\rm{x}}} \right)}}\)
\( = \frac{{\cos 2{\rm{x}} + \sin 2{\rm{x}}}}{{\cos 2{\rm{x}} - \sin 2{\rm{x}}}}\) (đpcm).