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Chapter 6: Problem 87

Show that $$ \tan ^{2}(2 x)=\frac{4\left(\cos ^{2} x-\cos ^{4} x\right)}{\left(2 \cos ^{2} x-1\right)^{2}} $$ for all numbers \(x\) except odd multiples of \(\frac{\pi}{4}\).

Short Answer

Expert verified
We use the double-angle formula for tangent, \(\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}\), and express tangent in terms of cosine, \(\tan(x) =\frac{\sin(x)}{\cos(x)}\). Then, we simplify the expression and square it: \(\tan^2(2x) = \frac{4\sin^2(x)\cos^2(x)}{(\cos^2(x)-\sin^2(x))^2}\). Using the identity \(\sin^2(x)=1-\cos^2(x)\), we substitute and simplify further, obtaining \(\tan^2(2x) = \frac{4\cos^2(x)-4\cos^4(x)}{(2\cos^2(x)-1)^2}\). Finally, we have shown that \[ \tan^2(2x) = \frac{4\left(\cos^2(x)-\cos^4(x)\right)}{(2\cos^2(x)-1)^2} \] for all numbers \(x\) except odd multiples of \(\frac{\pi}{4}\).

Step by step solution

01

Write down the double-angle formula for tangent

First, we need to write down the double-angle formula for tangent which is given by: \[ \tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)} \]
02

Express tangent in terms of cosine

Now express the tangent in terms of cosine by using the formula \[ \tan(x) =\frac{\sin(x)}{\cos(x)} \] We have: \[ \tan(2x) = \frac{2\frac{\sin(x)}{\cos(x)}}{1-\left(\frac{\sin(x)}{\cos(x)}\right)^2} \]
03

Simplify the expression and square both sides

Now, we can simplify the expression: \[ \tan(2x) = \frac{2\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)} \] Square both sides of the equation: \[ \tan^2(2x) = \frac{4\sin^2(x)\cos^2(x)}{(\cos^2(x)-\sin^2(x))^2} \]
04

Use the identity \(\sin^2(x)=1-\cos^2(x)\)

Now, we will use the identity \(\sin^2(x)=1-\cos^2(x)\). Replace \(\sin^2(x)\) in the numerator and denominator with this expression: \[ \tan^2(2x) = \frac{4\cos^2(x)(1-\cos^2(x))}{(\cos^2(x)-(1-\cos^2(x)))^2} \]
05

Simplify the expression further

Now, simplify the expression further: \[ \tan^2(2x) = \frac{4\cos^2(x)-4\cos^4(x)}{(2\cos^2(x)-1)^2} \] Therefore, we have shown that \[ \tan^2(2x) = \frac{4\left(\cos^2(x)-\cos^4(x)\right)}{(2\cos^2(x)-1)^2} \] for all numbers \(x\) except odd multiples of \(\frac{\pi}{4}\).

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