Problem 190 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \tan \theta=\frac{\sin \alpha \sin \beta}{\cos \alpha+\cos \beta}, \text { prove that one of the values of } \tan \frac{\theta}{2} \text { is } \tan \frac{\alpha}{2} \tan \frac{\beta}{2} \text { . } $$

Step-by-Step Solution

Answer

\(\tan \frac{\theta}{2} = \tan \frac{\alpha}{2} \tan \frac{\beta}{2}\) is one of the values of \(\tan \frac{\theta}{2}\) as required by the exercise.

Step 1: Expression of tan(θ) in terms of sin and cos

The given expression is \(\tan \theta=\frac{\sin \alpha \sin \beta}{\cos \alpha+\cos \beta}\).

Step 2: Use of half-angle identities

Next, express \(\sin\) and \(\cos\) in terms of half angles, using the relationships \(\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}\) and \(\cos x = 1 - 2 \sin^2 \frac{x}{2}\). We substitute these formulas into the original equation to get: \(\tan \theta = \frac{2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} \cdot 2 \sin \frac{\beta}{2} \cos \frac{\beta}{2}}{1 - 2 \sin^2 \frac{\alpha}{2} + 1 - 2 \sin^2 \frac{\beta}{2}}\). This simplifies to \(\tan \theta = \frac{4 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} \sin \frac{\beta}{2} \cos \frac{\beta}{2}}{2 - 2 \sin^2 \frac{\alpha}{2} - 2 \sin^2 \frac{\beta}{2}}\)

Step 3: Simplifying and rearranging the equation

On simplifying further, it gives \(\tan \theta = \frac{2 \sin \frac{\alpha}{2} \sin \frac{\beta}{2}}{1 - \sin^2 \frac{\alpha}{2} - \sin^2 \frac{\beta}{2}}\), which on separating the terms, gives \(\tan \theta = 2 \sin \frac{\alpha}{2} \sin \frac{\beta}{2} \cdot \frac{1}{1 - \sin^2 \frac{\alpha}{2} - \sin^2 \frac{\beta}{2}}\). The denominator can be rewritten using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\). Hence \(\tan \theta = 2 \sin \frac{\alpha}{2} \sin \frac{\beta}{2} \cdot \frac{1}{\cos^2 \frac{\alpha}{2} + \cos^2 \frac{\beta}{2}}\)

Step 4: Identification of tan(θ/2) in the expression

Observe closely and we can find \(\tan \frac{\alpha}{2} \tan \frac{\beta}{2}\) and \(\tan \theta\) in the rewritten equation, \(\tan \theta = \tan \frac{\alpha}{2} \tan \frac{\beta}{2} \cdot \frac{2}{1 - \tan^2 \frac{\alpha}{2} \tan^2 \frac{\beta}{2}}\). Divide both sides by \( \tan \frac{\alpha}{2} \tan \frac{\beta}{2}\) to isolate \(\tan \frac{\theta}{2}\) on one side of the equation to get \(\tan \frac{\theta}{2} = \tan \frac{\alpha}{2} \tan \frac{\beta}{2}\)

Key Concepts

Half-Angle Identities

In trigonometry, half-angle identities are a set of equations that allow us to express trigonometric functions of half-angles in terms of the full angle. These identities are particularly useful in integration, simplifying expressions, and solving trigonometric equations.

One common half-angle identity is for the sine function: \[\sin\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos(x)}{2}}\] This becomes useful when an expression involves \(\sin\left(\frac{\alpha}{2}\right)\) or \(\sin\left(\frac{\theta}{2}\right)\), as shown in the exercise where this identity helps express \(\sin\alpha\) and \(\sin\beta\) in terms of their half-angles. Similarly, for the cosine function, we use: \[\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \cos(x)}{2}}\] The use of the plus or minus sign depends on the quadrant in which the half-angle lies. These equations are derived from the angle sum and difference identities and are vital for transforming and simplifying trigonometric expressions.

Trigonometric Functions

Trigonometric functions are fundamental to the study of triangles, waves, and periodic phenomena. The basic trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function relates the angles of a right triangle to the ratios of two of its sides.

The sine function compares the length of the opposite side to the hypotenuse, the cosine function compares the adjacent side to the hypotenuse, and the tangent function is the ratio of the sine to the cosine of an angle. In the given exercise, \(\tan\theta\) is expressed in terms of the product of \(\sin\alpha\) and \(\sin\beta\), and with the help of half-angle identities, we transform this expression into a function of half-angles. This manipulation allows for the proof to advance since it simplifies the complex relationship between the tangents of the full angle and half-angles.

Angle Sum and Difference Identities

The angle sum and difference identities are another set of essential tools in trigonometry. These formulas express the sine, cosine, or tangent of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. For instance, the sine of the sum of two angles is given by: \[\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\] Similar identities exist for cosine and tangent. These identities not only help in solving complex trigonometric problems but also in proving other trigonometric equations like the one in our exercise. The proven identity for \(\tan\frac{\theta}{2}\), which relates to the product of the tangents of half of the angles \(\alpha\) and \(\beta\), involves rearranging the given expression using the angle sum and difference identities as part of the transformation process, thus showing their interconnectivity and importance in trigonometric proofs.