/*! 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 44 Verify the identity. $$\sec ^{... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 44

Verify the identity. $$\sec ^{2} y-\cot ^{2}\left(\frac{\pi}{2}-y\right)=1$$

Short Answer

Expert verified
After applying trigonometric properties, co-function identity, and Pythagorean identity, the original equation simplifies to 1=1, successfully verifying the given identity.

Step by step solution

01

Understand the Given Function

Recognize that the term \( \cot\left(\frac{\pi}{2}-y\right)\) can be re-written in terms of sine and cosine functions, making use of the fact that \( \cot(x)=\frac{\cos(x)}{\sin(x)} \)
02

Rewrite in Terms of Cosine and Sine

Rewrite the original equation as \( \sec^{2} y-\left(\frac{\cos\left(\frac{\pi}{2}-y\right)}{\sin\left(\frac{\pi}{2}-y\right)}\right)^{2}=1 \). Here, secant function \( \sec y = \frac{1}{\cos y} \) can also be converted to a function of cosine to facilitate further simplification.
03

Apply Co-Function Identity

The cosine of \( \frac{\pi}{2}-y \) is equal to the sine of y, and the sine of \( \frac{\pi}{2}-y \) is equal to the cosine of y. This stems from the Co-Function Identity in trigonometry. So, \( \left(\frac{\cos\left(\frac{\pi}{2}-y\right)}{\sin\left(\frac{\pi}{2}-y\right)}\right)^{2} \) turns into \( \left(\frac{\sin y}{\cos y}\right)^{2} \). Replace in the equation to get \( \frac{1}{\cos^{2} y}-\left(\frac{\sin y}{\cos y}\right)^{2}=1 \).
04

Denominator Unification

To simplify the equation, make the denominators the same for both terms on left side: \( \frac{1-\sin^{2} y}{\cos^{2} y}=1 \)
05

Apply Pythagorean Identity

Substitute using the Pythagorean Identity, \( 1-\sin^{2} y=\cos^{2} y \). So, \( \frac{1-\sin^{2} y}{\cos^{2} y} \) turns into \( \frac{\cos^{2} y}{\cos^{2} y} \).
06

Final Step and Verification

Therefore, \( \frac{\cos^{2} y}{\cos^{2} y} \) equals to 1. This verifies the original identity.

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