Problem 144 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \sin (\pi \cos x)=\cos (\pi \sin x), \text { prove that } \cos \left(x \pm \frac{\pi}{4}\right)=\frac{1}{2 \sqrt{2}} \text { . } $$

Step-by-Step Solution

Answer

The provided statement for proof \(\cos \left(x \pm \frac{\pi}{4}\right) = \frac{1}{2\sqrt{2}}\) is not correct.

Step 1: Analyze given equation

Begin by looking at the given equation: \(\sin (\pi \cos x)=\cos (\pi \sin x)\). This is true for all x, as it is a trigonometric identity derived using co-function identities. However, for simplicity, we can take \(x = \frac{\pi}{4} + n\pi\), where n is any integer, since \(\pi\cos(x)=\pi\sin(x)\) is true for these values of x.

Step 2: Substitute the simplified x in the equation to be proven

With the value of x from Step 1, substitute into \(\cos \left(x \pm \frac{\pi}{4}\right)\). This yields: \(\cos\left(\frac{\pi}{4} + n\pi \pm \frac{\pi}{4}\right)\). Simplify this to: \(\cos\left(n\pi + \frac{1}{2}\pi \right)\) or \(\cos\left(n\pi\right)\), depending on choosing + or -.

Step 3: Apply cosine fundamental identities

For any integer n, \(\cos(n\pi)=(-1)^n\), and, \(\cos\left(n\pi + \frac{1}{2}\pi \right)=0\). Since our objective is to prove \(\cos \left(x \pm \frac{\pi}{4}\right) = \frac{1}{2\sqrt{2}}\), the only possibility is \(\cos(n\pi)\).

Step 4: Set equation to be proven and solve

Now, we can setup the equation \((-1)^n = \frac{1}{2\sqrt{2}}\). Clearly, there is no integer n for which this equation holds. Therefore, the proof is not possible, leading to the conclusion that the provided statement for proving is incorrect.

Key Concepts

Cofunction Identities

Cofunction identities are key relationships between trigonometric functions that highlight how one function can be expressed in terms of another. These identities often involve complementary angles. Complementary angles are two angles whose measures add up to 90 degrees or \( \frac{\pi}{2} \) radians.
Cofunction identities can be defined for sine and cosine, tangent and cotangent, as well as secant and cosecant.

One of the most crucial cofunction identities is:

• \( \sin(\frac{\pi}{2} - x) = \cos(x) \)
• \( \cos(\frac{\pi}{2} - x) = \sin(x) \)

In the context of this problem, cofunction identities help explain the relationship between \( \sin(\pi \cos x) \) and \( \cos(\pi \sin x) \). They hold true due to the symmetrical nature of the sine and cosine functions, which are horizontally shifted versions of each other. Thus, understanding these identities is instrumental in manipulating trigonometric equations.

Angle Addition Formulas

Angle addition formulas play a vital role in trigonometry by allowing us to express the trigonometric functions of an angle sum or difference in terms of trigonometric functions of the individual angles. This means that we can easily solve problems involving compound angles.

The angle addition formulas for cosine are particularly useful here. They are:

• \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
• \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \)

By applying these formulas, the problem involves simplifying expressions like \( \cos(x \pm \frac{\pi}{4}) \). When expanded, these allow the compound angles to be broken down into simpler components. This simplification is critical in problem-solving as it makes the equations more manageable and easier to understand.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and need to be solved for the variable, usually the angle. Such equations can be straightforward or quite complex, as with the current problem

When dealing with trigonometric equations:

• Simplification is often necessary, sometimes using identities.
• Considering special angle solutions like integer multiples of \( \pi \).
• Leveraging periodic properties of functions.

In the problem, the equation \( \sin(\pi \cos x) = \cos(\pi \sin x) \) illustrates a situation where solutions are found using the symmetric nature of sine and cosine functions. The approach involves testing specific values of \( x \) by considering their periodic nature, which sometimes reveals solutions or contradictions in such equations. This can lead to insights about whether a specific identity or statement about the equation holds true. Understanding these concepts is fundamental in solving and evaluating trigonometric equations effectively.