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Chapter 6: Problem 85

Show that $$ (\cos \theta+\sin \theta)^{2}(\cos \theta-\sin \theta)^{2}+\sin ^{2}(2 \theta)=1 $$ for all angles \(\theta\).

Short Answer

Expert verified
We expand and simplify the given expression step-by-step: 1. Expand the terms inside the parentheses using the formulas \((a+b)^2=a^2+b^2+2ab\) and \((a-b)^2=a^2+b^2-2ab\). 2. Use the Pythagorean trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\) to simplify. 3. Expand the product of the expressions and simplify. 4. Rewrite the sine term using the double angle formula for sine, \(\sin(2\theta) = 2\cos \theta \sin \theta\). 5. Combine the expressions and simplify to show that the expression is equal to 1 for all angles 胃.

Step by step solution

01

Expand the terms inside the parantheses

Using the formula \((a + b)^2 = a^2 + b^2 + 2ab\) and \((a - b)^2 = a^2 + b^2 - 2ab\), we can expand the expressions inside the parantheses. \[ (\cos \theta+\sin \theta)^{2}(\cos \theta-\sin \theta)^{2} = [\cos^2 \theta + \sin^2 \theta + 2\cos \theta \sin \theta] [\cos^2 \theta + \sin^2 \theta - 2\cos \theta \sin \theta] \]
02

Use Pythagorean Identity

Using the Pythagorean trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can simplify the expression further. \[ [\cos^2 \theta + \sin^2 \theta + 2\cos \theta \sin \theta] [\cos^2 \theta + \sin^2 \theta - 2\cos \theta \sin \theta] = (1 + 2\cos \theta \sin \theta)(1 - 2\cos \theta \sin \theta) \]
03

Expand the product of the expressions and simplify

Expand the product and simplify the expression. \[ (1 + 2\cos \theta \sin \theta)(1 - 2\cos \theta \sin \theta) = 1 - 4\cos^2 \theta \sin^2 \theta \]
04

Simplify the sin term as well

Using the double angle formula for sine, \(\sin(2\theta) = 2\cos \theta \sin \theta\), we can rewrite the given sine term. \[ \sin^2(2\theta) = (2\cos \theta \sin \theta)^{2}= 4\cos^2 \theta \sin^2 \theta \]
05

Combine the expressions

Add the two expressions and simplify. \[ (1 - 4\cos^2 \theta \sin^2 \theta) + 4\cos^2 \theta \sin^2 \theta = 1 \] We have shown that the given expression is indeed equal to 1 for all angles 胃.

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