Question

prove that 1+cos105+cos165/sin105+sin375=0​

Answer 1

Given :  1 + Cos105°  + Cos165° /Sin 105° + Sin 375°  = 0

To find : Prove

Solution:

1 + Cos105°  + Cos165° /Sin 105° + Sin 375°

Cos α =  - Cos(180° - α)

Sinα =  Sin(180° - α)

Sin (α+ 360°) = Sinα

= 1  -Cos75°  -Cos15°/ Sin75°  + Sin 15°

Cos α = Sin (90° - α )

Sin α = Cos (90° - α )

= 1  -Sin15°  -Cos15°/ Cos15°  + Sin 15°

= 1 - Sin 15°  - 1  + Sin 15°

= 0

QED

Hence Proved

1 + Cos105°  + (Cos165° /Sin 105°) + Sin 375°    = 0

if  1 + (Cos105°  + Cos165°) /(Sin 105° + Sin 375°)

= 1 + ( -Sin15° - Sin75°)/(Sin75° + Sin15°)

= 1  - (Sin75° + Sin15°)/(Sin75° + Sin15°)

= 1 - 1

= 0

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