Question

Proving Identities

Verify the identity$\frac{\sin x+\cos x}{\sec x+\csc x}=\sin x\cos x$

Short answer

The expression$\frac{\sin x+\cos x}{\sec x+\csc x}=\sin x\cos x$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression$\frac{\sin x+\cos x}{\sec x+\csc x}=\sin x\cos x$

Step 2. Concept used.

To verify the given identity, split it into LHS and RHS pieces. Then, one by one, simplify them. When both results are the same, the supplied expression is called an identity.

Step 3. Calculation.

Now, simplify LHS of $\frac{\sin x+\cos x}{\sec x+\csc x}=\sin x\cos x$:

Write the expression in terms of $\sin x\&\cos x$,

$\begin{array}{l}\frac{\sin x+\cos x}{\sec x+\csc x}=\frac{\sin x+\cos x}{\frac{1}{\cos x}+\frac{1}{\sin x}}\\ =\frac{\sin x+\cos x}{\frac{\sin x+\cos x}{\sin x\times \cos x}}\\ =\sin x\cos x\end{array}$

RHS of the equation is$\sin x\times \cos x$

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