/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Verify the identity. \(\frac{\... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 35

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

Short Answer

Expert verified
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

01

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 \(\sec y = \frac{1}{\cos y}\). \( \frac{\sin y}{\csc y}+\frac{\cos y}{\sec y}=\frac{\sin y}{\frac{1}{\sin y}}+\frac{\cos y}{\frac{1}{\cos y}} \)
02

Simplify the fractions

Now, we will simplify the fractions by multiplying the numerators and denominators by their reciprocal function: \( = \sin y \cdot \frac{\sin y}{1} + \cos y \cdot \frac{\cos y}{1} \)
03

Distribute the products

Next, distribute the products to get: \( = \sin^2 y + \cos^2 y \)
04

Apply the Pythagorean identity

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

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