Question

Demonstrati ca: a) $\frac{cos 7+sin 7}{cos7-sin7}$ =tg 52
b) $\frac{cos 13+sin 13}{cos 13-sin 13}$ =ctg 32
c) $\frac{sin53-cos37}{sin53+cos37}$ =tg k$\pi$ , k∈Z
d) $\frac{1}{sin84}- \frac{1}{sin12} + \frac{1}{sin24}+ \frac{1}{sin48} = ctg(2k+1) \frac{ \pi }{2}$ , k∈Z

Answer 1

E₁ = [ sin83 +sin7 ]/ [ sin83 -sin7 ] 
cos 7 = sin (90 -7)  =sin 83
transformam suma (diferenta ) in produs 
E₁= [ 2· sin ( 83+7) /2   ·cos( 83 -7)/2]   /  [ 2 ·sin ( 83-7)/2 · cos ( 83+7)/2  ]
=[ 2· sin 45 · cos 38 ]  /[ 2 · sin 38 ·cos 45 ] 
simplificam , cos 45= sin 45 
= cos 38 / sin38 = ctg 38 = tg 52
E₂ = [ cos 13 + cos 77] / [ cos 13 - cos 77 ] =
=[ 2 cos 45 cos 32] / [ -2 sin45 sin ( - 32 )] simplificam   impara  sin (-x) = - sinx
=cos32 / sin32  = ctg 32