Question

How do you factor the expression and use the fundamental identities to simplify $1-2{{\cos }^{2}}x+{{\cos }^{4}}x$?

Answer 1

Hint: We will use the basic trigonometric identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ to solve the given expression. By substituting the value from the identity and by splitting the middle term and taking common terms out we will get the factors of the given expression.

Complete step by step solution:
We have been given an expression $1-2{{\cos }^{2}}x+{{\cos }^{4}}x$.
We have to find the factors of the given expression.
Now, we can rewrite the given equation as
$\Rightarrow 1-{{\cos }^{2}}x-{{\cos }^{2}}x+{{\cos }^{4}}x$
Now, taking the common terms out we will get
\[\begin{align}
  & \Rightarrow 1-{{\cos }^{2}}x-{{\cos }^{2}}x+{{\cos }^{4}}x \\
 & \Rightarrow 1\left( 1-{{\cos }^{2}}x \right)-{{\cos }^{2}}x\left( 1-{{\cos }^{2}}x \right) \\
\end{align}\]
Again taking common factors out we will get
$\Rightarrow \left( 1-{{\cos }^{2}}x \right)\left( 1-{{\cos }^{2}}x \right)$
Hence we get the factors of the given equation as $\left( 1-{{\cos }^{2}}x \right)$$\left( 1-{{\cos }^{2}}x \right)$.
Now, we know that ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ or ${{\sin }^{2}}x=1-{{\cos }^{2}}x$
Substituting the values in the obtained factors we will get
\[\Rightarrow {{\sin }^{2}}x{{\sin }^{2}}x\]
Simplifying the above obtained equation we will get
\[\Rightarrow {{\sin }^{4}}x\]

Note: To avoid mistakes alternatively students can solve the given expression by assuming ${{\cos }^{2}}x=y$ then substituting the values we will get
$\Rightarrow 1-2y+{{y}^{2}}$
Rearranging the terms we will get
$\Rightarrow {{y}^{2}}-2y+1$
The above obtained equation is a quadratic equation and we can easily factor the equation. After factoring again we will substitute the value $y={{\cos }^{2}}x$.
We can factor the above equation by splitting the middle term then we will get
\[\Rightarrow {{y}^{2}}-y-y+1\]
Now, taking common terms out we will get
$\begin{align}
  & \Rightarrow y\left( y-1 \right)-1\left( y-1 \right) \\
 & \Rightarrow \left( y-1 \right)\left( y-1 \right) \\
\end{align}$
Now, again substituting the value $y={{\cos }^{2}}x$ we will get
$\Rightarrow \left( {{\cos }^{2}}x-1 \right)\left( {{\cos }^{2}}x-1 \right)$
Now, we know that ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ or ${{\sin }^{2}}x=1-{{\cos }^{2}}x$
Substituting the values in the obtained factors we will get
\[\Rightarrow {{\sin }^{2}}x{{\sin }^{2}}x\]
Simplifying the above obtained equation we will get
\[\Rightarrow {{\sin }^{4}}x\]
Hence we get the factors of the given expression.