Question

докажите тождество:

$frac{1 - cos 2t + sin 2t }{1 + cos 2t + sin 2t}$ * tg ( $frac{ pi }{2}$ - t ) = 1

Answer 1

1) 1 - cos2t + sin2t = cos^2t + sin^2t - cos^2t + sin^2t +2sintcost = 

= 2sin^2t + 2sintcost = 2sint(sint + cost)

 

2) 1 + cos2t + sin2t = cos^2t + sin^2t + cos^2t - sin^2t +2sintcost = 

= 2cos^2t + 2sintcost = 2cost(cost + sint)

 

3) tg(П/2 - t) = ctgt

 

4) 2sint(sint + cost)*ctgt/[2cost(cost + sint)] = tgt*ctgt = 1