Understanding the Trigonometric Equation Problem

The problem asks us to find a pair of trigonometric values for 'a' and 'b' that satisfies the given equation: \(a^2 = b^2 + (ab)^2\).

We are provided with four options, each giving specific trigonometric expressions for 'a' and 'b'. We need to substitute these values into the equation and check which pair makes the equation true.

Testing the Given Trigonometric Options

Let's examine each option by substituting the values of 'a' and 'b' into the equation \(a^2 = b^2 + (ab)^2\) and simplifying both sides.

Option 1: a = sin x, b = cot x

Substitute \(a = \sin x\) and \(b = \cot x\) into the equation:

Left Hand Side (LHS): \(a^2 = (\sin x)^2 = \sin^2 x\)

Right Hand Side (RHS): \(b^2 + (ab)^2 = (\cot x)^2 + (\sin x \cdot \cot x)^2\)

We know that \(\cot x = \frac{\cos x}{\sin x}\). Substitute this into the RHS:

RHS: \(\cot^2 x + \left(\sin x \cdot \frac{\cos x}{\sin x}\right)^2\)

RHS: \(\cot^2 x + (\cos x)^2\)

RHS: \(\cot^2 x + \cos^2 x\)

Comparing LHS and RHS: Is \(\sin^2 x = \cot^2 x + \cos^2 x\)? This is generally false.

So, option 1 does not satisfy the equation.

Option 2: a = cos x, b = tan x

Substitute \(a = \cos x\) and \(b = \tan x\) into the equation:

Left Hand Side (LHS): \(a^2 = (\cos x)^2 = \cos^2 x\)

Right Hand Side (RHS): \(b^2 + (ab)^2 = (\tan x)^2 + (\cos x \cdot \tan x)^2\)

We know that \(\tan x = \frac{\sin x}{\cos x}\). Substitute this into the RHS:

RHS: \(\tan^2 x + \left(\cos x \cdot \frac{\sin x}{\cos x}\right)^2\)

RHS: \(\tan^2 x + (\sin x)^2\)

RHS: \(\tan^2 x + \sin^2 x\)

Comparing LHS and RHS: Is \(\cos^2 x = \tan^2 x + \sin^2 x\)? This is generally false.

So, option 2 does not satisfy the equation.

Option 3: a = cot x, b = cos x

Substitute \(a = \cot x\) and \(b = \cos x\) into the equation:

Left Hand Side (LHS): \(a^2 = (\cot x)^2 = \cot^2 x\)

Right Hand Side (RHS): \(b^2 + (ab)^2 = (\cos x)^2 + (\cot x \cdot \cos x)^2\)

We know that \(\cot x = \frac{\cos x}{\sin x}\). Substitute this into the RHS:

RHS: \(\cos^2 x + \left(\frac{\cos x}{\sin x} \cdot \cos x\right)^2\)

RHS: \(\cos^2 x + \left(\frac{\cos^2 x}{\sin x}\right)^2\)

RHS: \(\cos^2 x + \frac{\cos^4 x}{\sin^2 x}\)

To add these terms, find a common denominator, which is \(\sin^2 x\):

RHS: \(\frac{\cos^2 x \sin^2 x}{\sin^2 x} + \frac{\cos^4 x}{\sin^2 x}\)

RHS: \(\frac{\cos^2 x \sin^2 x + \cos^4 x}{\sin^2 x}\)

Factor out \(\cos^2 x\) from the numerator:

RHS: \(\frac{\cos^2 x (\sin^2 x + \cos^2 x)}{\sin^2 x}\)

Using the fundamental trigonometric identity \(\sin^2 x + \cos^2 x = 1\):

RHS: \(\frac{\cos^2 x (1)}{\sin^2 x}\)

RHS: \(\frac{\cos^2 x}{\sin^2 x}\)

Using the identity \(\cot x = \frac{\cos x}{\sin x}\):

RHS: \((\cot x)^2 = \cot^2 x\)

Comparing LHS and RHS: We found LHS = \(\cot^2 x\) and RHS = \(\cot^2 x\). Since LHS = RHS, this option satisfies the equation.

Option 4: a = sin x, b = tan x

Substitute \(a = \sin x\) and \(b = \tan x\) into the equation:

Left Hand Side (LHS): \(a^2 = (\sin x)^2 = \sin^2 x\)

Right Hand Side (RHS): \(b^2 + (ab)^2 = (\tan x)^2 + (\sin x \cdot \tan x)^2\)

RHS: \(\tan^2 x + \sin^2 x \tan^2 x\)

Factor out \(\tan^2 x\):

RHS: \(\tan^2 x (1 + \sin^2 x)\)

Comparing LHS and RHS: Is \(\sin^2 x = \tan^2 x (1 + \sin^2 x)\)? This is generally false.

So, option 4 does not satisfy the equation.

Conclusion on Trigonometric Values for a and b

Based on the step-by-step verification, the pair of values \(a = \cot x\) and \(b = \cos x\) is the only option among the given choices that satisfies the trigonometric equation \(a^2 = b^2 + (ab)^2\).

Revision Table: Key Trigonometric Identities

Here are some fundamental trigonometric identities used in solving this problem:

Additional Information on Solving Trigonometric Equations

Solving trigonometric equations often involves using identities to simplify the expression or equation. The goal is usually to express everything in terms of a single trigonometric function or to reach a known identity.

In this problem, we tested specific values, but for general trigonometric equations, you might need to:

• Use reciprocal identities (\(\csc x = 1/\sin x\), \(\sec x = 1/\cos x\), \(\cot x = 1/\tan x\)).
• Use quotient identities (\(\tan x = \sin x/\cos x\), \(\cot x = \cos x/\sin x\)).
• Use Pythagorean identities (\(\sin^2 x + \cos^2 x = 1\), \(1 + \tan^2 x = \sec^2 x\), \(1 + \cot^2 x = \csc^2 x\)).
• Factor trigonometric expressions.
• Find common denominators when adding or subtracting fractions with trigonometric terms.

Practicing with different identities helps in simplifying and solving complex trigonometric problems like verifying if certain values satisfy an equation.