Question

If A+B+C=180 degree then prove that tanB.tanC + tanC.tanA + tanA.tanB = 1 + secA.secB.secC

Answer 1

LHS =

  = [ sinA SInB CosC + SinA SinC CosB + SinB SinC CosA ] / cosA CosB CosC
  = [ Sin A * (Sin B Cos C + CosB SinC) + Sin B Sin C CosA] sec A sec B sec C
  = [ sin A Sin (B+C) + SIn B SIn C Cos A ] sec A sec B sec C
  = [ sin² A + Sin B SIn C Cos A ] sec A sec B sec C
  = [ 1 - Cos² A + sin B Sin C Cos A ] sec A sec B sec C
  = sec A Sec B sec C  + (Sin B Sin C - Cos A) sec B Sec C
 = sec A sec B sec C + [Sin B Sin C + Cos (B+C) ]sec B sec C
   = sec A sec B sec C  + [ Cos B Cos C ] * sec B Sec C
  = sec A   sec B sec C  + 1