Problem 39 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\left(\cos ^{2} x-1\right)\left(\tan ^{2} x+1\right)=-\tan ^{2} x$$

Step-by-Step Solution

Answer

$$\left(\cos ^{2} x-1\right)\left(\tan ^{2} x+1\right)=-\tan ^{2} x$$ Answer: Yes, the given equation is an identity.

Step 1: Rewrite LHS using identities

We will start by rewriting the LHS using the identities mentioned earlier. The LHS is given as: $$\left(\cos^2 x - 1\right) \left(\tan^2 x + 1\right)$$ Using identity 1, we know that $$\cos^2 x = 1 - \sin^2 x$$, so we can write the LHS as: $$\left(1 - \sin^2 x - 1\right) \left(\tan^2 x + 1\right)$$ Now, using identity 2, we know that $$\tan^2 x + 1 = \sec^2 x = \frac{1}{\cos^2 x}$$, so we can write the LHS as: $$- \sin^2 x \left(\frac{1}{\cos^2 x}\right)$$ Finally, since $$\tan x = \frac{\sin x}{\cos x}$$, we can write the LHS as: $$-\tan^2 x$$

Step 2: Compare the LHS and RHS

Now that we have simplified the LHS, we can compare it to the RHS of the given equation. The equation is: $$\left(\cos ^{2} x-1\right)\left(\tan ^{2} x+1\right)=-\tan ^{2} x$$ After simplification, we found that the LHS is equal to $$-\tan^2 x$$. The RHS is also equal to $$-\tan^2 x$$. Therefore, the given equation is an identity, as we have proved that the LHS and RHS are equal.

Key Concepts

Cosine Identity

When we talk about trigonometric identities, the cosine identity is one of the fundamental ones to understand. The basic form of the cosine identity is given by \(cos^2 x = 1 - \sin^2 x\). This comes from the Pythagorean identity which states \(sin^2 x + \cos^2 x = 1\). By rearranging this equation, we get the cosine identity.
While this might seem simple, it is a powerful tool for simplifying expressions and proving equations. For example, in our exercise, we utilized this identity to transform the expression \(\cos^2 x - 1\) into \(-(1 - \cos^2 x) = - \sin^2 x\). This step is crucial because it allows us to express everything in terms of sine and thus simplifies the handling of the equation altogether.
Understanding and identifying where you can apply this identity can make solving trigonometric equations much easier. It showcases how manipulating one side of an equation through identities can result in significant simplifications.

Tangent Identity

The tangent identity links tangent and secant functions to each other through the relation \(tan^2 x + 1 = \sec^2 x\). This identity stems from rearranging the Pythagorean identity \(1 + \tan^2 x = \sec^2 x\), indicating a fundamental relationship between these trigonometric functions.
In terms of our exercise, the tangent identity helped to rewrite \(\tan^2 x + 1\) as \(\sec^2 x\). This representation allows additional simplifications since \(\sec^2 x = \frac{1}{\cos^2 x}\). This transformation played a crucial role in matching the left side of the given equation with the right side, ultimately proving the expression as an identity.

• Simplification: Using tangent identity helps when expressions involve \(\tan(x)\) or \(\sec(x)\), allowing you to rewrite them in simpler forms.
• Adaptability: The identity shows flexibility and can be used in various equation types involving tangent and secant functions.

Mastering the use of the tangent identity is key in solving complex trigonometric equations effectively.

Secant Identity

The secant identity is directly derived from the rearranged Pythagorean theorem. As noted in the tangent identity, \(\sec^2 x = 1 + \tan^2 x\). Secant is the reciprocal of cosine, i.e., \(\sec x = \frac{1}{\cos x}\). Therefore, \(\sec^2 x = \frac{1}{\cos^2 x}\).
This identity was crucial in our exercise because it allowed us to link sine and cosine through secant. Rewriting \(\tan^2 x + 1\) as \(\sec^2 x\) helped simplify the left part of the equation. By using secant identity, we demonstrated how the terms on the left-hand side mirror those on the right-hand side, confirming the equation is an identity.

• Transformation: Convert secant expressions into cosine based on definitions and identities, which helps in transformation and simplification of trigonometric expressions.
• Analytical Tool: Using the secant identity is vital in expressing problems using reciprocal functions which simplifies analyses.

Understanding the secant identity is essential for dealing with intricate trigonometric problems, especially when explicitly including reciprocal functions.