Question

Prove the following Trigonometric expression:
$\cos 4x=1-8{{\sin }^{2}}x{{\cos }^{2}}x$

Answer 1

Hint:To prove the given expression, apply the identity of $\cos 2\theta $ on $\cos 4x$ then you will get$1-2{{\sin }^{2}}2x$. Now, apply $\sin 2x$ as $2\sin x\cos x$ in the expression $1-2{{\sin }^{2}}2x$ and hence, simplify the expression to make it equal to R.H.S of the given expression.

Complete step-by-step answer:
The equation that we have to prove is:
$\cos 4x=1-8{{\sin }^{2}}x{{\cos }^{2}}x$
We are going to apply the identity of $\cos 2\theta $ on $\cos 4x$ as:
$\begin{align}
  & \cos 2\theta =1-2{{\sin }^{2}}\theta \\
 & \cos 4x=1-2{{\sin }^{2}}2x \\
\end{align}$
We know that $\sin 2x=2\sin x\cos x$. So, applying this value of $\sin 2x$ in the above equation we get,
$\begin{align}
  & \cos 4x=1-2{{\left( 2\sin x\cos x \right)}^{2}} \\
 & \Rightarrow \cos 4x=1-8{{\sin }^{2}}x{{\cos }^{2}}x \\
\end{align}$
From the above simplification, we have got $\cos 4x$ as $1-8{{\sin }^{2}}x{{\cos }^{2}}x$ which is equal to R.H.S of the given expression.
Hence, we have proved that $\cos 4x=1-8{{\sin }^{2}}x{{\cos }^{2}}x$.

Note: In spite of solving the L.H.S of the given equation:
$\cos 4x=1-8{{\sin }^{2}}x{{\cos }^{2}}x$
We can also resolve R.H.S to L.H.S. Let’s check out how we are going to do this.
The R.H.S of given equation is:
$1-8{{\sin }^{2}}x{{\cos }^{2}}x$
Rewriting the above expression as:
$1-2{{(2\sin x\cos x)}^{2}}$
Now, we know that $\sin 2x=2\sin x\cos x$ so we can write $2\sin x\cos x$ in the above equation as $\sin 2x$.
$1-2{{\left( \sin 2x \right)}^{2}}$
From the trigonometric double angles identity we know that $\cos 2\theta =1-2{{\sin }^{2}}\theta $. In the above expression the value of θ is equal to 2x so substituting the value of θ in $\cos 2\theta =1-2{{\sin }^{2}}\theta $ we get, $\cos 4x=1-2{{\sin }^{2}}2x$. Hence, we can write $1-2{{\sin }^{2}}2x$ as $\cos 4x$.
From the above simplification the R.H.S has come out as $\cos 4x$ which is equal to L.H.S.
Hence, we have proved L.H.S = R.H.S of the given expression by resolving R.H.S to L.H.S.