Rút gọn biểu thức lượng giác: $tan5^{0}.tan55^{0}.tan65^{0}.tan75^{0}$

Question

Rút gọn biểu thức lượng giác:

$\tan5^{0}.\tan55^{0}.\tan65^{0}.\tan75^{0}$

score 1 · Answer 1

Ta có:

$\tan5^o.\tan55^o.\tan65^o.\tan 75^o$

$=\dfrac{\sin5^o.\sin 55^o. \sin 65^o}{\cos5^o.\cos55^o.\cos65^o}.\tan75^o$

$=\dfrac{\sin5^o.(\cos10^o-\cos120^o)}{\cos5^o.(\cos10^o+\cos120^o)}.\tan75^o$

$=\dfrac{2\sin5^o\cos10^o+\sin5^o}{2\cos5^o\cos10^o-\cos5^o}.\tan75^o$

$=\dfrac{\sin15^o-\sin5^o+\sin5^o}{\cos15^o+\cos5^o-\cos5^o}.\tan75^o$

$=\dfrac{\sin15^o}{\cos15^o}.\tan75^o$

$=\tan15^o.\cot15^o=1$

1 Đáp án