Toán 10 Lượng giác

1) $\sin a + \cos a = m$
$\implies (\sin a + \cos a)^2 = m^2$
$\implies \sin^2 a + \cos^2 a + 2 \sin a \cos a = m^2$
$\implies 1 + 2 A = m^2$
$\implies A = \dfrac{m^2-1}{2}$

2) $B = \sin a - \cos a$
$\implies B^2 = (\sin a - \cos a)^2$
$\implies B^2 = \sin^2 a + \cos^2 a - 2 \sin a \cos a$
$\implies B^2 = 1 - 2 \cdot \dfrac{m^2-1}{2} = 1 - (m^2 - 1) = 2 - m^2$
$\implies B = \sqrt{2 - m^2}$

3) $C = \sin^4 a + \cos^4 a$
$\implies C = (\sin^2 a + \cos^2 a)^2 - 2 \sin^2 a \cos^2 a$
$\implies C = 1 - 2 \cdot \left(\dfrac{m^2-1}{2}\right)^2$ (Có thể tự rút gọn tiếp nếu muốn)

4) $D = \tan^2 a + \cot^2 a$
$\implies D = \dfrac{1}{\cos^2 a} - 1 + \dfrac{1}{\sin^2 a} - 1$
$\implies D = \dfrac{\sin^2 a + \cos^2 a}{\sin^2 a \cos^2 a} - 2$
$\implies D = \dfrac{1}{\left(\dfrac{m^2-1}{2}\right)^2} - 2 = \dfrac{4}{(m^2-1)^2}- 2$