If x sin3|+ y cos3|=sin|cos| and xsin|=ycos|, prove x2+y2=1.

Question is ---        X  sin³β + Ycos³β =sinβcosβ    and,   X sinβ =Y cosβ , have to be proved,           X²  + Y² =1 . 

                         Or    Xsinβ sin²β + Ycos³β = sinβcosβ .

                                  Y cosβsin²β + Ycos²β = sinβcosβ  ( as  X sinβ  = Y cosβ)  .

                               Y cosβ ( sin²β + cos²β ) = sinβ cosβ . (as sin²β + cos²β  = 1).

                       So Y cosβ = sinβ cosβ . or  Y= sinβ .   now  Xsinβ =Ycos β   ( put Y=sinβ) . then Xsinβ  =sinβcosβ .  so X = cosβ.

  So   X²   + Y²  = cos²β  + sin²β  =1   proved   answer

Question is      X²  + Y² =1 . 

                       So Y cosβ = sinβ cosβ . or  Y= sinβ .   now  Xsinβ =Ycos β   ( put Y=sinβ) . then Xsinβ  =sinβcosβ .  so X = cosβ.

  So   X²   + Y²  = cos²β  + sin²β  =1   proved   answer

Question is ---        X  sin³β + Ycos³β =sinβcosβ    and,   X sinβ =Y cosβ , have to be proved,           X²  + Y² =1 . 

                         Or    Xsinβ sin²β + Ycos³β = sinβcosβ .

                                  Y cosβsin²β + Ycos²β = sinβcosβ  ( as  X sinβ  = Y cosβ)  .

                               Y cosβ ( sin²β + cos²β ) = sinβ cosβ . (as sin²β + cos²β  = 1).

                       So Y cosβ = sinβ cosβ . or  Y= sinβ .   now  Xsinβ =Ycos β   ( put Y=sinβ) . then Xsinβ  =sinβcosβ .  so X = cosβ.

  So   X²   + Y²  = cos²β  + sin²β  =1   proved   answer

Question is ---        X  sin³β + Ycos³β =sinβcosβ    and,   X sinβ =Y cosβ , have to be proved,           X²  + Y² =1 . 

                         Or    Xsinβ sin²β + Ycos³β = sinβcosβ .

                                  Y cosβsin²β + Ycos²β = sinβcosβ  ( as  X sinβ  = Y cosβ)  .

                               Y cosβ ( sin²β + cos²β ) = sinβ cosβ . (as sin²β + cos²β  = 1).

                       So Y cosβ = sinβ cosβ . or  Y= sinβ .   now  Xsinβ =Ycos β   ( put Y=sinβ) . then Xsinβ  =sinβcosβ .  so X = cosβ.

  So   X²   + Y²  = cos²β  + sin²β  =1   proved   answer

Question is ---                  X²  + Y² =1 .