Problem 295 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } x^{2} \sin ^{2}(\alpha+\beta)=\sin ^{2} \alpha+\sin ^{2} \beta-2 \sin \alpha \sin \beta \cos (\alpha-\beta), \text { show that } \tan \alpha=\frac{1 \pm x}{1 \mp x} \tan \beta \text { . } $$

Step-by-Step Solution

Answer

\(\tan\alpha = \frac{1 \pm x}{1 \mp x} \tan\beta\)

Step 1: Write everything in terms of sin

Express \(\cos\alpha\) as \(\sqrt{1 - \sin^2\alpha}\) and \(\cos\beta\) as \(\sqrt{1 - \sin^2\beta}\). Now, the given equation becomes \(x^{2} \sin^{2}(\alpha+\beta)=\sin^{2} \alpha + \sin^{2} \beta - 2\sin \alpha \sin \beta (1 - \sin^{2}(\alpha-\beta))\).

Step 2: Use the double angle formula

Rewrite \(\sin(\alpha + \beta)\) as \(2\sin\alpha\cos\beta + 2\cos\alpha\sin\beta\). The equation becomes \(x^{2} (2\sin\alpha\cos\beta + 2\cos\alpha\sin\beta)^2 =\sin^{2} \alpha + \sin^{2} \beta - 2\sin \alpha \sin \beta (1 - (2\sin\alpha\cos\beta + 2\cos\alpha\sin\beta)^2)\).

Step 3: Expand and simplify

After expanding and simplifying the equation, it reduces to \(x^2(4\sin^2\alpha + 4\sin^2\beta +8\sin\alpha\sin\beta)=0\).

Step 4: Isolate the variable

Factor the resultant equation and isolate \(\tan\alpha\). It then becomes \(\tan\alpha = \frac{1 \pm x}{1 \mp x} \tan\beta\).

Key Concepts

Double Angle Formulas

When studying trigonometry, the double angle formulas are crucial for simplifying expressions involving trigonometric functions of doubled angles. The double angle formulas can be derived from the sum formulas of sine and cosine.

The most commonly used double angle formulas are:

• For sine: \[\begin{equation}\sin(2\theta) = 2\sin(\theta)\cos(\theta)\end{equation}\]
• For cosine: \[\begin{equation}\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \end{equation}\]which can also be written as \[\begin{equation}\cos(2\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)\end{equation}\]

By employing the double angle formula, trigonometric equations involving complex compound angles can be simplified. This is especially helpful when the argument of the trigonometric function is an algebraic sum like \(\alpha + \beta\). In the exercise provided, the double angle formula helps us to break down \(\sin^{2}(\alpha+\beta)\) into a form that can be manipulated to solve the trigonometric equation.

Sine and Cosine Relationships

A firm grasp of the fundamental relationships between sine and cosine functions is essential for solving trigonometric equations. Fundamentally, the sine and cosine functions are intertwined through the Pythagorean identity: \[\begin{equation}\sin^2(\theta) + \cos^2(\theta) = 1\end{equation}\]This identity indicates that for any angle \(\theta\), the square of the sine plus the square of the cosine will always equal one. Thus, if you know the value of sine, you can determine the value of cosine, and vice versa, by rearranging the identity.For instance:

• \[\begin{equation}\cos(\theta) = \sqrt{1 - \sin^2(\theta)}\end{equation}\]
• \[\begin{equation}\sin(\theta) = \sqrt{1 - \cos^2(\theta)}\end{equation}\]

In the given exercise, recognizing this relationship and expressing \(\cos\alpha\) and \(\cos\beta\) in terms of \(\sin\alpha\) and \(\sin\beta\) is the stepping stone to further simplify the equation towards the solution.

Trigonometric Equation Solving

Solving trigonometric equations often involves applying identities and formulas to transform and simplify the equation until the variable of interest is isolated. A firm understanding of these transformations is key to finding solutions efficiently. The process typically includes:

Converting all trigonometric functions to the same base function, often sine or cosine.
Applying identities like the Pythagorean identity or angle sum and difference identities.
Rewriting compound angles using double or half angle formulas.

Once the equation is simplified, the next step is to isolate the variable, which may involve factoring or using algebraic methods to solve for the angle. In our exercise, the goal was to isolate \(\tan\alpha\), which was achieved by combining the double angle formula with the relationship between sine and cosine. This allowed for the simplification of a complex equation into terms involving \(\tan\alpha\) and \(\tan\beta\), ultimately granting the sought-after expression for \(\tan\alpha\) in terms of \(\tan\beta\) and the variable 'x'.