Question

Proving Identities

Verify the identity$\frac{\sin A}{1-\cos A}-\cot A=\csc A$

Short answer

The expression$\frac{\sin A}{1-\cos A}-\cot A=\csc A$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression$\frac{\sin A}{1-\cos A}-\cot A=\csc A$

Step 2. Concept used.

Split the phrase into two pieces, LHS and RHS, to check if it is an identity or not. After that, make both of them simpler. If the outcomes are the same, the expression is said to be an identity.

Step 3. Calculation.

Now, simplify LHS of $\frac{\sin A}{1-\cos A}-\cot A=\csc A$:

$$\begin{gathered} \frac{\sin A}{1-\cos A}-\cot A=\frac{{\sin }^2{\displaystyle }A-(1-\cos A)\cos A}{(1-\cos A)\sin A} \\ \begin{array}{l}=\frac{{\sin }^{2}A-\left(\cos A-{\cos }^{2}A\right)}{(1-\cos A)\sin A}\\ =\frac{{\sin }^{2}A-\cos A+{\cos }^{2}A}{(1-\cos A)\sin A}\\ =\frac{{\sin }^{2}A+{\cos }^{2}A-\cos A}{(1-\cos A)\sin A}\\ =\frac{1-\cos A}{(1-\cos A)\sin A}\\ =\frac{1-\cos A}{1-\cos A}\times \frac{1}{\sin A}\\ =\frac{1}{\sin A}=\csc A\end{array} \end{gathered}$$

RHS of the equation is$\csc A$

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