Question

Verify the identity. \(\frac{\sin y}{\csc y}+\frac{\cos y}{\sec y}=1\)

Short answer

To verify the identity \(\frac{\sin y}{\csc y}+\frac{\cos y}{\sec y}=1\), we first express it in terms of sine and cosine functions: \(\frac{\sin y}{\frac{1}{\sin y}}+\frac{\cos y}{\frac{1}{\cos y}}\). Simplify the fractions by multiplying the numerators and denominators by their reciprocal function to get \(\sin^2 y + \cos^2 y\). Finally, apply the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\), confirming that the identity holds.

Step-by-step solution

Express in terms of sine and cosine

We will begin by replacing the cosecant and secant functions with their respective reciprocal identities: \(\csc y = \frac{1}{\sin y}\) and \(\sec y = \frac{1}{\cos y}\). \( \frac{\sin y}{\csc y}+\frac{\cos y}{\sec y}=\frac{\sin y}{\frac{1}{\sin y}}+\frac{\cos y}{\frac{1}{\cos y}} \)

Simplify the fractions

Now, we will simplify the fractions by multiplying the numerators and denominators by their reciprocal function: \( = \sin y \cdot \frac{\sin y}{1} + \cos y \cdot \frac{\cos y}{1} \)

Distribute the products

Next, distribute the products to get: \( = \sin^2 y + \cos^2 y \)

Apply the Pythagorean identity

Now, apply the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\): \( = 1 \) We have successfully verified the given identity: \( \frac{\sin y}{\csc y}+\frac{\cos y}{\sec y}=1 \)