Chứng minh rằng:

a) \(\cos a - \sin a = \sqrt 2 \cos \left( {a + \frac{\pi }{4}} \right)\);

b) \(\sin a + \sqrt 3 \cos a = 2\sin \left( {a + \frac{\pi }{3}} \right)\).

Lời giải

Vì \(\frac{\pi }{4}\) < x < \(\frac{\pi }{2}\) nên sin x > 0, cos x > 0. Áp dụng công thức hạ bậc, ta có

\({\sin ^2}x = \frac{{1 - \cos 2x}}{2} = \frac{{1 - \left( { - \frac{4}{5}} \right)}}{2} = \frac{9}{{10}}\) ⇒ sin x = \(\frac{3}{{\sqrt {10} }}\).

\({\cos ^2}x = \frac{{1 + \cos 2x}}{2} = \frac{{1 + \left( { - \frac{4}{5}} \right)}}{2} = \frac{1}{{10}}\) ⇒ cos x = \(\frac{1}{{\sqrt {10} }}\).

Theo công thức nhân đôi, ta có sin 2x = 2 sin x cos x = \(2.\frac{3}{{\sqrt {10} }}.\frac{1}{{\sqrt {10} }} = \frac{6}{{10}} = \frac{3}{5}\).

Theo công thức cộng, ta có

\(\sin \left( {x + \frac{\pi }{3}} \right) = \sin x\cos \frac{\pi }{3} + \cos x\sin \frac{\pi }{3} = \frac{3}{{\sqrt {10} }}.\frac{1}{2} + \frac{1}{{\sqrt {10} }}.\frac{{\sqrt 3 }}{2} = \frac{{3 + \sqrt 3 }}{{2\sqrt {10} }}\).

\[\cos \left( {2x - \frac{\pi }{4}} \right) = \cos 2x\cos \frac{\pi }{4} + \sin 2x\sin \frac{\pi }{4} = \left( { - \frac{4}{5}} \right).\frac{{\sqrt 2 }}{2} + \frac{3}{5}.\frac{{\sqrt 2 }}{2} = - \frac{{\sqrt 2 }}{{10}}\].

Lời giải

\(VT = \sin A + \sin B + \sin C\)\( = 2\sin \frac{{A + B}}{2}\cos \frac{{A - B}}{2} + 2\sin \frac{C}{2}\cos \frac{C}{2}\).

Mặt khác, trong tam giác ABC, ta có A + B + C = π nên \(\frac{{A + B}}{2} = \frac{\pi }{2} - \frac{C}{2}\).

Từ đó suy ra: \(\sin \frac{{A + B}}{2} = \cos \frac{C}{2},\,\sin \frac{C}{2} = \cos \frac{{A + B}}{2}\).

Vậy \(VT = 2\sin \frac{{A + B}}{2}\cos \frac{{A - B}}{2} + 2\sin \frac{C}{2}\cos \frac{C}{2}\)

\( = 2\cos \frac{C}{2}\cos \frac{{A - B}}{2} + 2\cos \frac{{A + B}}{2}\cos \frac{C}{2}\)

\( = 2\cos \frac{C}{2}\left( {\cos \frac{{A - B}}{2} + \cos \frac{{A + B}}{2}} \right)\)

\( = 2\cos \frac{C}{2}.2\cos \frac{{\frac{{A - B}}{2} + \frac{{A + B}}{2}}}{2}\cos \frac{{\frac{{A - B}}{2} - \frac{{A + B}}{2}}}{2}\)

\( = 4\cos \frac{C}{2}\cos \frac{A}{2}\cos \left( { - \frac{B}{2}} \right)\)

\( = 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2} = VP\) (điều phải chứng minh).