Question

Proving Identities

Verify the identity$\csc x-\sin x=\cos x\cot x$

Short answer

The expression$\csc x-\sin x=\cos x\cot x$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression$\csc x-\sin x=\cos x\cot x$

Step 2. Concept used.

Time will be simplified in two ways. The first time simplify the LHS component of the equation, and the second time simplify the RHS component of the expression. The expression is an identity if both results are the same.

Step 3. Calculation.

Now, simplify LHS of $\csc x-\sin x=\cos x\cot x$:

Formula used,

$$\begin{gathered} =\csc x-\sin x \\ \because \csc x=\frac{1}{\sin x} \\ \because {\sin }^{2}x+{\cos }^{2}x=1 \\ \because \frac{\cos x}{\sin x}=\cot x \end{gathered}$$

$$\begin{gathered} =\frac{1}{\sin {\displaystyle }x}-\sin x \\ =\frac{1-{\sin }^{2}x}{\sin {\displaystyle }x} \\ =\frac{{\cos }^2{\displaystyle }x}{\sin {\displaystyle }x} \\ =\frac{\cos {\displaystyle }x}{\sin {\displaystyle }x}\times \cos x \\ =\cos x\cot x \end{gathered}$$

RHS of the equation is$\cos x\cot x$

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