Question

Вычислите; tg3-tg 48)/(1+tg 3*ctg 42)

Answer 1

$\frac{\tan 3-\tan 48}{1+\tan 3*\cot 42}=\frac{\frac{\sin 3}{\cos 3}-\frac{\sin 48}{\cos 48}}{1+\frac{\sin 3}{\cos 3}*\frac{\cos 42}{\sin 42}}=\\ =\frac{\sin 3 \cos 48-\sin 48\cos 3}{\cos 3 \sin 42+\sin 3\cos 42}*\frac{\cos 3 \sin 42}{\cos 3 \cos 48}=\\ \frac{\sin(48-3)}{\sin(3+42)}*\frac{\sin 42}{\cos 48}=\frac{\sin 45}{\sin 45} *\frac{\sin 42}{\cos 48}= \frac{\sin (45-3)}{\cos (45+3)}=\\ = \frac{\sin 45\cos3-\cos 45 \sin 3}{\cos 45\cos3-\sin 45 \sin 3}=$

 

$= \frac{\frac{\sqrt{2}}{2}(\cos 3 - \sin 3)}{\frac{\sqrt{2}}{2}(\cos 3 - \sin 3)} = 1$