Question

If $ x = a\cos \theta - b\sin \theta $ and $ y = a\sin \theta + b\cos \theta $ then prove that $ {x^2} + {y^2} = {a^2} + {b^2} $

Answer 1

Hint: To prove the given expression we need to find the square of each x and y separately then add them and simplify till RHS will not be achieved. We make use appropriate trigonometric identities.

Complete step-by-step answer:
Given $ x = a\cos \theta - b\sin \theta $ and $ y = a\sin \theta + b\cos \theta $ we have to prove $ {x^2} + {y^2} = {a^2} + {b^2} $
So in LHS it is given that we need to find $ {x^2} + {y^2} $
So taking the value of x and squaring both side we get,
 $ {x^2} = {a^2}{\cos ^2}\theta + {b^2}{\sin ^2}\theta - 2ab\sin \theta \cos \theta $
Similarly to find the value of $ {y^2} $ we will squaring the value of y on both side we get,
Now to get the value of LHS just add the value of $ {x^2} $ and $ {y^2} $
On adding we get,
 $ \Rightarrow {x^2} + {y^2} = {a^2}{\cos ^2}\theta + {b^2}{\sin ^2}\theta - 2ab\sin \theta \cos \theta + {a^2}{\sin ^2}\theta + {b^2}{\cos ^2}\theta + 2ab\sin \theta \cos \theta $
Here the 2ab term will get cancelled out
So finally we get,
 $ \Rightarrow {x^2} + {y^2} = {a^2}{\cos ^2}\theta + {b^2}{\sin ^2}\theta + {a^2}{\sin ^2}\theta + {b^2}{\cos ^2}\theta $
 $ \Rightarrow {x^2} + {y^2} = {a^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) + {b^2}\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right) $
Now taking $ {a^2} $ and $ {b^2} $ we get
 $ \Rightarrow {x^2} + {y^2} = {a^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) + {b^2}\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right) $
We know the value of $ {\sin ^2}\theta + {\cos ^2}\theta = 1 $
Applying this result on the above we get
 $ \Rightarrow {x^2} + {y^2} = {a^2} + {b^2} $
LHS=RHS
Hence it is proved.
So, the correct answer is “ $ {x^2} + {y^2} = {a^2} + {b^2} $ ”.

Note: Students also initiate from RHS to LHS means first find the value $ {a^2} $ and $ {b^2} $ then put it in RHS and solved such that you will get LHS. Student can go for this way but this is quite difficult and opposite process. So try to go from LHS to RHS.