Bài tập 4 trang 154 SGK Đại số 10

Câu a:

Ta có \(VT = \frac{{{\rm{cos(a - b)}}}}{{{\rm{cos}}(a + b)}}\frac{{cosacosb + sinasinb}}{{cosacosb - sinasinb}}\)

\( = \frac{{\frac{{cosacosb}}{{sinasinb}} + 1}}{{\frac{{cosacosb}}{{sinasinb}} - 1}} = \frac{{cota.cotb + 1}}{{cota.cotb - 1}}\)

 Câu b:

Ta có: \(\sin (a + b).\sin (a - b)\)

\( = \left[ {(sina.cosb{\rm{ }} + {\rm{ }}cosa.sinb).(sina.cosb{\rm{ }} - {\rm{ }}cosa.sina)} \right]\)

\( = {\left( {sina.cosb} \right)^2}-{\rm{ }}{\left( {cosa.sinb} \right)^2} = {\rm{ }}si{n^2}a(1{\rm{ }}-{\rm{ }}si{n^2}b){\rm{ }}--{\rm{ }}(1{\rm{ }}-{\rm{ }}si{n^2}a)si{n^2}b\)

\( = {\sin ^2}a.c{\rm{o}}{{\rm{s}}^2}b = (1 - c{\rm{o}}{{\rm{s}}^2}a) - (1 - c{\rm{o}}{{\rm{s}}^2}a)\)

\( = c{\rm{o}}{{\rm{s}}^2}b - c{\rm{o}}{{\rm{s}}^2}a\) (đpcm)

Câu c:

Ta có: \({\rm{cos(a + b)}}{\rm{.sin(a - b)}}\)

\( = (c{\rm{osa}}{\rm{.cosb - }}\sin a.\sin b)(cosa.cosb + sina.sinb)\)

\( = {\cos ^2}a{\cos ^2}b - {\sin ^2}a{\sin ^2}b\)

\( = {\cos ^2}a(1 - {\sin ^2}b) - {\sin ^2}b(1 - {\cos ^2}a) = {\cos ^2}a - {\sin ^2}b\)

\( = (1 - {\sin ^2}a) - (1 - {\cos ^2}b) = {\cos ^2}b - {\sin ^2}a\) (đpcm)

-- Mod Toán 10 HỌC247