Problem 35 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\sec (x+y)+\sec (x-y)=2 \sec x\), where \(x, y \in\left(0, \frac{\pi}{2}\right)\), then prove that \(\cos x=\sqrt{2} \cos \left(\frac{y}{2}\right)\).

Step-by-Step Solution

Answer

The proof that \(\cos x = \sqrt{2} \cos (y/2)\) was completed using trigonometric identities and transformations, a critical aspect of which was accounting for the fact that x and y are in the same quadrant, restricting them to positive values.

Step 1: Express the secants as cosine functions

First, we convert the secant functions into cosine functions since it's simpler to manipulate. Recall that \(\sec\theta = 1/\cos\theta\). So, \(\sec (x+y) + \sec (x-y) = 1/\cos(x+y) + 1/\cos(x-y) = 2/\cos x\).

Step 2: Cross Multiply

After obtaining the expression in terms of cosine, cross multiply to eliminate the fractions: \( => \cos x \cos(x-y) + \cos x \cos(x+y) = 2 \cos(x+y) \cos(x-y)\)

Step 3: Use the Identity of Cosine

Now, apply the addition and subtraction formulas: \(\cos a \cos b = (1/2)(\cos(a-b) + \cos(a+b))\). This gives us: \(2 \cos^2 x = \cos(2y) + 1\)

Step 4: Express the equation in terms of x

Rewrite the equation above: \(\cos^2 x = 1/2 (1+\cos 2y)\). Remembering the properties of the cosine function, we can rewrite \(1+\cos 2y\) as \(2\cos^2 y\), as \(\cos 2y = 2\cos^2 y - 1\). This simplifies the equation to \(\cos^2 x = \cos^2 y\)

Step 5: Simplify the Equation

The equation \(\cos^2 x = \cos^2 y\) can be further simplified to \(\cos x = ±\cos y\). We know that both x and y are in the same quadrant \(0 < x, y < \frac{\pi}{2}\), so the cosine of x will be positive if and only if the cosine of y is positive. Therefore, we eliminate the negative sign. \(\cos x = \cos y\)

Step 6: Final Proof

Use the half-angle identity \(\cos y = \sqrt{2} \cos (y/2)\), substitute into \(\cos x = \cos y\) to get \(\cos x = \sqrt{2} \cos (y/2)\). This completes the proof.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function: \( \sec \theta = 1/\cos \theta \). This function is useful when solving trigonometric equations that involve division by cosine, especially when the cosine value is known to be non-zero.

When working with secant, it's crucial to remember that it shares the same domain restrictions as the cosine function, meaning it is undefined whenever \( \cos \theta = 0 \).

The secant function can also provide unique insights when solving equations because it extends the range beyond [-1, 1] as seen with cosine, making it handy in special trigonometric identities and expressions.

Cosine Function

The cosine function, \( \cos \theta \), is one of the primary functions in trigonometry. It describes a basic wave-like pattern and is useful in calculating parts of right-angle triangles, among other geometric applications.

In terms of a unit circle, \( \cos \theta \) gives the x-coordinate of a point located at an angle \( \theta \) from the positive x-axis. Due to its periodic nature, cosine helps in dealing with cycles and oscillations in both geometry and physics.

Important properties include its even nature, meaning that \( \cos(-\theta) = \cos(\theta) \), and its periodicity, where \( \cos(\theta + 2\pi) = \cos(\theta) \). These properties are often applied in simplifying trigonometric equations.

Half-Angle Identity

The half-angle identities are crucial tools in trigonometric simplifications and proofs. They allow us to express trigonometric functions of half-angles in terms of square roots of expressions involving non-half-angle functions.

The half-angle identity for cosine states that:

• \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)

Depending on the quadrant the angle \( \theta/2 \) lies in, the positive or negative sign is determined by the cosine function's value in that quadrant.

In our exercise, the half-angle identity assists in transitioning from \( \cos y \) to \( \cos \frac{y}{2} \), essential for concluding that \( \cos x = \sqrt{2} \cos \left( \frac{y}{2} \right) \). This derivation is a powerful demonstration of trigonometric transformation to show equivalencies.

Addition and Subtraction Formulas

The addition and subtraction formulas are valuable tools for simplifying expressions involving trigonometric functions of combined angles.

The cosine addition and subtraction formulas are:

• Cosine Addition: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• Cosine Subtraction: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

These formulas provide a precise method to express trigonometric functions of sums and differences of angles in terms of products of sine and cosine of simpler angles.

In the part of our solution where we derived \( \cos x \cos(x-y) + \cos x \cos(x+y) \), we leveraged these formulas to break down the expressions into manageable components.