Question

Proving Identities

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

Short answer

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

Step-by-step solution

Step 1. Given information.

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

Step 2. Concept used.

To answer this sort of question, divide the expression into two parts: LHS and RHS. After that, simplify both portions. If the answers are equivalent after the simplification, the supplied expression is an identity.

Step 3. Calculation.

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

$$\begin{gathered} \cot x=\frac{\cos x}{\sin x} \\ =(\cot x-\csc x)(\cos x+1) \\ =\left(\frac{\cos x}{\sin x}-\frac{1}{\sin x}\right)(\cos x+1) \\ =\left(\frac{\cos x-1}{\sin x}\right)(\cos x+1) \\ =\frac{(\cos x-1)\cdot (\cos x+1)}{\sin x} \\ =\frac{\left({\cos }^{2}x-1\right)}{\sin x} \\ =\frac{-{\sin }^{2}x}{\sin x} \\ =-\sin x \end{gathered}$$

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

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