Question

establish the identity.
\frac{sec\theta - csc\theta}{sec\thetacsc\theta}=sin\theta - cos\theta

write the left side as a difference of two quotients.
\frac{square}{sec\thetacsc\theta}-\frac{square}{sec\thetacsc\theta}

Explanation:

Step1: Recall reciprocal identities

We know that $\sec\theta=\frac{1}{\cos\theta}$ and $\csc\theta = \frac{1}{\sin\theta}$.
The left - hand side of the identity is $\frac{\sec\theta-\csc\theta}{\sec\theta\csc\theta}$.
Writing it as a difference of two quotients: $\frac{\sec\theta}{\sec\theta\csc\theta}-\frac{\csc\theta}{\sec\theta\csc\theta}$.

Step2: Simplify each quotient

For the first quotient $\frac{\sec\theta}{\sec\theta\csc\theta}$, cancel out $\sec\theta$ in the numerator and denominator, we get $\frac{1}{\csc\theta}$. Since $\csc\theta=\frac{1}{\sin\theta}$, then $\frac{1}{\csc\theta}=\sin\theta$.
For the second quotient $\frac{\csc\theta}{\sec\theta\csc\theta}$, cancel out $\csc\theta$ in the numerator and denominator, we get $\frac{1}{\sec\theta}$. Since $\sec\theta=\frac{1}{\cos\theta}$, then $\frac{1}{\sec\theta}=\cos\theta$.
So $\frac{\sec\theta}{\sec\theta\csc\theta}-\frac{\csc\theta}{\sec\theta\csc\theta}=\sin\theta - \cos\theta$, which is the right - hand side of the given identity.

Answer:

The identity $\frac{\sec\theta-\csc\theta}{\sec\theta\csc\theta}=\sin\theta - \cos\theta$ is established.