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

Math magician

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... read more

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 .

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