Elimination of angle

Elimination of angle 
Elimination of angle (\(\theta\))  by using trigonometric identities
The process of transforming trigonometric equation into an Algebric equation using trigonometric identities is called elimination of \(\theta\).  Here ‘ \(\theta\) ‘ is an angle which is defined
Illustration -1 :
    Eliminate ‘\(\theta\)‘ from the equations \(x = a\,\sin \theta ,\,\,y = a\cos \theta \Rightarrow \sin \theta = \frac{x}{a},\,\,\cos \theta = \frac{y}{a}\)
Solution :
We know \({\sin ^2}\theta + \cos {\theta ^2} = 1(\rlap{--} V\theta \in R)\)
\(\Rightarrow {\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{a}} \right)^2} = 1\) 
\(\Rightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1\)
\(\Rightarrow {x^2} + {y^2} = {a^2}\)
Illustration -2 :
    Eliminate ‘\(\theta \)’ from the equation \(x = \sec \theta + \tan \theta ,\,\,\,y = \sec \theta - \tan \theta \).
Solution :
We know 
 \(\Rightarrow (\sec \theta + \tan \theta )(\sec \theta - \tan \theta ) = 1\)  [(a+b)² = a² +b² +2ab ]
 \( \Rightarrow \,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y\,\,\,\,\,\,\,\,\,\,\,\, = 1\)
  \(\Rightarrow xy = 1\)
Illustration -3 :
    Eliminate ‘\(\theta\)’ from the trigonometric equation \(a = x\cos \theta + y\sin \theta \)  and \(b = x\sin \theta - y\cos \theta \)
Solution :
    Squaring and adding the given equations 
    \(\left[ \begin{gathered} \because {(a + b)^2} = {a^2} + {b^2} + 2ab \hfill \\ {(a - b)^2} = {a^2} + {b^2} - 2ab \hfill \\ \end{gathered} \right]\)
\({a^2} + {b^2} = {(x\cos \theta + y\sin \theta )^2} + {(x\sin \theta - y\cos \theta )^2}\) 
\(= \,\,{x^2}({\sin ^2}\theta + {\cos ^2}\theta ) + {y^2}({\sin ^2}\theta + {\cos ^2}\theta ) + 2xy\sin \theta \cos \theta - \)\(2xy\sin \theta \cos \theta \) \(\left[ {\because {{\sin }^2}\theta \, + {{\cos }^2}\theta = 1} \right]\)
\(= {x^2}(1) + {y^2}(1)\)
\(= {x^2}(1) + {y^2}(1)\)            
   a²+b²=x²+y²