Problem 256 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \begin{aligned} &\text { If } \cos \theta=\cos \alpha \cos \beta \text { and } \cos \phi=\cos \gamma \cos \beta \text { where } \cos \beta \neq 0 \text { and } \tan \frac{p}{2}=\tan \frac{\sigma}{2} \tan \frac{\varphi}{2}, \text { prove that }\\\ &\sin ^{2} \beta=(\sec \alpha-1)(\sec \gamma-1) \end{aligned} $$

Step-by-Step Solution

Answer

The proof involves using the trigonometric identities and simplifying the equations. The important part is to transform the equations to the form given in the statement to be proved, which eventually leads to \(sin^2\beta = (\sec\alpha -1)(\sec\gamma -1)\).

Step 1: Expression of \(\cos\theta\) and \(\cos\phi\) in terms of \(\cos\beta\)

Given \(\cos\theta=\cos\alpha\cos\beta\) and \(\cos\phi=\cos\gamma\cos\beta\), implies that \(\cos\theta/\cos\alpha=\cos\beta\) and \(\cos\phi/\cos\gamma=\cos\beta\).

Step 2: Use the property \(\cos^2\beta=1-\sin^2\beta\)

Next, radar the given equations using the identity \(\cos^2\beta=1-\sin^2\beta\). We get \(1-\sin^2\beta=\cos\theta/\cos\alpha\) and \(1-\sin^2\beta=\cos\phi/\cos\gamma\).

Step 3: Express \(sin^2\beta\)

Express \(sin^2\beta\) from the above two equations, we find \(sin^2\beta=1-\cos\theta/\cos\alpha\) and \(sin^2\beta=1-\cos\phi/\cos\gamma\).

Step 4: Replace \(\cos\theta\) and \(\cos\phi\)

Note that \(\cos\alpha = 1/\sec\alpha\) and \(\cos\gamma = 1/\sec\gamma\). We substitute these into the previous equations and find \(sin^2\beta=\sec\alpha - \cos\theta\) and \(sin^2\beta=\sec\gamma - \cos\phi\).

Step 5: Equate and prove the identity

Since both expressions equal to \(sin^2\beta\), we can equate them to find out the final result: \(\sec\alpha - \cos\theta = \sec\gamma - \cos\phi\). This simplifies to the required identity \(sin^2\beta = (\sec\alpha -1)(\sec\gamma -1)\)

Key Concepts

Cosine Rule

The Cosine Rule, also known as the Law of Cosines, is a fundamental principle in trigonometry that relates the sides and angles of a triangle. It generalizes the Pythagorean theorem to non-right triangles. The rule is usually written as:

• \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

This formula helps in calculating an unknown side of a triangle when two sides and the included angle are known. A crucial aspect of the Cosine Rule is its ability to find an angle when all sides of the triangle are known:

• \[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]

This rule is extremely useful in solving problems in geometric and trigonometric contexts, particularly when dealing with non-right triangles. It can help simplify complex calculations involving angles and sides, making it a vital tool in both theoretical and applied mathematics.

Half Angle Formulas

The Half Angle Formulas are essential identities in trigonometry that allow us to find the trigonometric functions of half-angles given the whole angle. These formulas are very useful when you need to solve problems involving angular equations or when simplifying expressions. The Half Angle Formulas for sine, cosine, and tangent are:

• \[ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \]
• \[ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \]
• \[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \]

The plus/minus sign depends on the quadrant in which the angle \(\theta/2\) resides. These formulas are particularly significant in calculus and other analytical techniques where integration or simplification requires these identities.

Secant Function

The Secant Function is one of the six fundamental trigonometric functions, defined as the reciprocal of the cosine function. It is expressed as:

• \[ \sec(\theta) = \frac{1}{\cos(\theta)}\]

In geometric terms, secant refers to a line that intersects a circle at two points. In trigonometry, understanding secant is crucial as it provides a way to solve equations and inequalities involving cosine. The secant function has similar periodic properties as the cosine function but it does not exist for angles where cosine is zero, since division by zero is undefined. Its graph consists of branches extending towards infinity at the asymptotes determined by the zeros of the cosine function. The secant function is used in various mathematical contexts, from solving trigonometrical equations to analyzing waveforms and oscillations in physics.