Question

Eliminate x, where $p{{\sin }^{3}}x+q{{\cos }^{3}}x=\sin x.\cos x$ and $p\sin x-q\cos x=0$ .

Answer 1

Hint:
1) Substitute the value of $p\sin x$ from the second equation into the first equation and solve for q.
2) Find the values of p and q in terms of $\sin x$ and/or $\cos x$ .
3) Make use of the fact that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ .

Complete step by step solution:
It is given that:
 $p{{\sin }^{3}}x+q{{\cos }^{3}}x=\sin x.\cos x$ ... (1)
 $p\sin x-q\cos x=0$ ... (2)
Using equation (2), we have:
 $p\sin x=q\cos x$ ... (3)
Substituting equation (3) in equation (1), we get:
 $q\cos x{{\sin }^{2}}x+q{{\cos }^{3}}x=\sin x.\cos x$
⇒ $q\cos x\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)=\sin x.\cos x$
⇒ $q\cos x=\sin x.\cos x$
⇒ $q=\sin x$ ... (4)
Using equations (4) and (3), we get:
 $p\sin x=\sin x\cos x$
⇒ $p=\cos x$ ... (5)
Finally, by squaring equations (4) and (5) and adding them together, we get:
 ${{p}^{2}}+{{q}^{2}}={{\sin }^{2}}x+{{\cos }^{2}}x$

⇒ ${{p}^{2}}+{{q}^{2}}=1$ , which is the required answer.

Note:
Eliminating 'x' from two or more equations means that the equations are combined logically into a single equation so that it remains valid and 'x' does not appear in this new equation.
Elimination of a variable is used in converting parametric form to cartesian form.
If there are more unknowns than the number of equations, or more specifically, for an under-determined system of equations, one variable can be eliminated, usually by substitution.