Question

Proving Identities

Verify the identity$2{\cos }^{2}x-1=1-2{\sin }^{2}x$

Short answer

The expression$2{\cos }^{2}x-1=1-2{\sin }^{2}x$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression$2{\cos }^{2}x-1=1-2{\sin }^{2}x$

Step 2. Concept used.

Divide the equation into LHS and RHS parts to solve this sort of problems. After that, both portions are simplified. If the answers are equivalent after the simplification, the expression is an identity.

Step 3. Calculation.

Now, simplify LHS of $2{\cos }^{2}x-1=1-2{\sin }^{2}x$:

$\begin{array}{l}{\cos }^{2}x=1-{\sin }^{2}x\\ \Rightarrow 2\left(1-{\sin }^{2}x\right)-1\\ \Rightarrow 2-2{\sin }^{2}x-1\\ \Rightarrow 1-2{\sin }^{2}x\end{array}$

RHS of the equation is$1-2{\sin }^{2}x$

Both RHS and LHS are equal. Hence it is an identity