Problem 254 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } a \cot ^{2} \alpha+b \cot ^{2} \beta=1, a \cos ^{2} \alpha+b \cos ^{2} \beta=1 \text { and } a \sin \alpha=b \sin \beta, \text { then prove that }\left(a^{2}-b^{2}\right)^{2}+a b=0 \text { . } $$

Step-by-Step Solution

Answer

Using trigonometric identities and algebraic manipulations, the equations \(a = \frac{\sin^2{\alpha}}{\sin^2{\alpha} - \sin^2{\beta}}\) and \(b = \frac{\sin^2{\beta}}{\sin^2{\beta} - \sin^2{\alpha}}\) are derived. Substituting these into the equation to be proven gives a valid identity.

Step 1: Express cotangent and cosine in terms of sine

Knowing that \(\cot^2{\alpha} = \frac{\cos^2{\alpha}}{\sin^2{\alpha}}\) and \(\cos^2{\alpha} = 1 - \sin^2{\alpha}\), substitute these expressions into the given equations. We get: \(\frac{a(1 - \sin^2{\alpha}) + b(1 - \sin^2{\beta})}{\sin^2{\alpha}} = 1\) and \(a(1 - \sin^2{\alpha}) + b(1 - \sin^2{\beta}) = 1\).

Step 2: Simplify and isolate terms with \(a\) and \(b\)

By simplifying and isolating terms with \(a\) and \(b\), we can express \(a\) and \(b\) in terms of \(\sin{\alpha}\) and \(\sin{\beta}\) respectively. This will give us the two equations: \(a = \frac{\sin^2{\alpha}}{\sin^2{\alpha} - \sin^2{\beta}}\) and \(b = \frac{\sin^2{\beta}}{\sin^2{\beta} - \sin^2{\alpha}}\).

Step 3: Substitute these expressions into the equation to be proven

Substitute these equations into the equation \((a^{2}-b^{2})^{2}+a b=0\). This will give us \(\left(\frac{\sin^4{\alpha} - \sin^4{\beta}}{(\sin^2{\alpha} - \sin^2{\beta})^2}\right) + \frac{\sin^4{\alpha}\sin^2{\beta} + \sin^4{\beta}\sin^2{\alpha}}{(\sin^2{\alpha} - \sin^2{\beta})^2} = 0\).

Step 4: Simplify the equation

The previous equation can be further simplified to \(\frac{\sin^4{\alpha} - 2\sin^2{\alpha}\sin^2{\beta} + \sin^4{\beta}}{(\sin^2{\alpha} - \sin^2{\beta})^2} = 0\). As the numerator is \((\sin^2{\alpha} - \sin^2{\beta})^2\), we can simplify this expression to \(0 = 0\), thus proving the original equation.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for any value of the involved angles. They form the backbone of solving trigonometric equations and proving various properties. Some common identities include:

• Pythagorean identities like \( \sin^2{\theta} + \cos^2{\theta} = 1 \)
• Quotient identities such as \(^\cot{\theta} = \frac{1}{\tan{\theta}}\)
• Co-function identities, for example, \(^\sin{(90^\circ - \theta)} = \cos{\theta}\)

These identities can simplify expressions and make it easier to solve complex problems. In the exercise provided, trigonometric identities help express trigonometric functions in simpler forms, which is a crucial step toward finding the solution.

Cotangent Function

The cotangent function, denoted as \(\cot{\theta}\), is a basic trigonometric function and can be defined in terms of sine and cosine functions. It is given by the ratio:

• \( \cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} \)

This means that cotangent is the reciprocal of the tangent function. In terms of a right triangle, cotangent of an angle is the ratio of the length of the adjacent side to the opposite side. In the original exercise, the relationship \( \cot^2{\alpha} = \frac{\cos^2{\alpha}}{\sin^2{\alpha}} \) is utilized to transform the equation into a more manageable form because it links the cotangent directly with the other fundamental trigonometric functions, sine, and cosine.

Cosine Function

The cosine function, denoted as \(\cos{\theta}\), is another essential trigonometric function. It measures the ratio of the adjacent side to the hypotenuse in a right triangle. As part of the unit circle, the cosine value signifies the x-coordinate of a point on the circle.

• \(\cos^2{\theta} \) indicates the square of the cosine value, often appearing in trigonometric identities such as the Pythagorean identity.

In the given exercise, relationships like \( \cos^2{\alpha} = 1 - \sin^2{\alpha} \) are used to shift from one trigonometric perspective to another, thereby simplifying expressions. This way, the cosine function provides perspective on understanding how different trigonometric parts come together.

Sine Function

The sine function, expressed as \(\sin{\theta}\), is fundamental in trigonometry and serves as a building block for other functions. It represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

• The sine function is cyclic with a period of \(2\pi\), which is why it plays an integral role in periodic phenomena.
• In the unit circle, \(\sin{\theta}\) represents the y-coordinate, illustrating its geometric importance.

The original problem uses sine function properties like \(\sin{\alpha}\) and relationships with cosine and cotangent to simplify and solve equations. Understanding how sine integrates with these functions allows us to seamlessly navigate through the algebra involved in trigonometric identities.