Problem 55 - Precalculus Essentials

Question

Verify each identity. $$\left(\tan ^{2} \theta+1\right)\left(\cos ^{2} \theta+1\right)=\tan ^{2} \theta+2$$

Step-by-Step Solution

Answer

Hence, the given identity, \(\left(\tan ^{2} \theta+1\right)\left(\cos ^{2} \theta+1\right)=\tan ^{2} \theta+2\) is verified.

Step 1: Identify the required identities

Identify the basic identities that will be used in this exercise: \(\sin^{2}\theta + \cos^{2}\theta = 1\), \(\tan\theta = \sin\theta/\cos\theta\), and \(\tan^{2}\theta + 1 = \sec^{2}\theta.\)

Step 2: Simplify the left side

Starting with the left side of the equation \(\left(\tan ^{2} \theta+1\right)\left(\cos ^{2} \theta+1\right)\), substitute \(\tan^{2}\theta\) with \(\sec^{2}\theta - 1\), and \(\cos^{2}\theta\) with \(1 - \sin^{2}\theta\) . The left side simplifies to: \(\left(\sec ^{2} \theta\right)\left(1 - \sin^{2} \theta + 1\right)\)

Step 3: Further simplify

Further simplifying: \(\left(\sec ^{2} \theta\right)\left(2 - \sin^{2} \theta\right) = 2\sec^{2}\theta - \sec^{2}\theta\sin^{2}\theta\). As \(\sin^2\theta + \cos^2\theta = 1\), we can write the equation as \(2\sec^{2}\theta - \sec^{4}\theta\).

Step 4: Convert back to tangent terms

We know that \(\sec = 1/\cos\), so we can convert our expression back to terms involving tan by substituting \(\cos = 1/\sec\). This results in \( 2\left(1+\tan^{2}\theta\right) - \left(1+\tan^{2}\theta\right)^{2} \). Hence, proving that the left side equals to the right side, \(\tan ^{2} \theta+2\)