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

Question

$$ 1+\cos ^{2} 2 \theta=2\left(\cos ^{4} \theta+\sin ^{4} \theta\right) $$

Step-by-Step Solution

Answer

The simplification process starting from the identity \(1+\cos^2 2\theta=2(\cos^4\theta+\sin^4\theta)\), leads to the result \(2\sin^2\theta - 2\sin^4\theta = 0\). This confirms the equality of the original identity.

Step 1: Simplify left side of the equation

The left side of the equation is already simple, so it stays as it is. It is \(1+\cos^2 2\theta\).

Step 2: Simplify right side of the equation

Use the identity \(\cos^2\theta\) with \(1-\sin ^{2}\theta\) to replace cos terms in the right side. So, \(2(\cos^4\theta + \sin^4\theta)\) becomes \(2((1-\sin^2\theta)^2 + \sin^4\theta)\).

Step 3: Expand and simplify

Simplify \(2((1-\sin^2\theta)^2 + \sin^4\theta)\), which becomes \(2\sin^4\theta - 4\sin^2\theta + 2\).

Step 4: Rearrange the equation

Rearrange the equation \(1+\cos^2 2\theta = 2\sin^4\theta - 4\sin^2\theta + 2\) to \(\cos^2(2\theta) + 4\sin^2\theta - 2\sin^4\theta = 1\).

Step 5: Use the double angle formula

For \(\cos^2(2\theta)\), use the double angle formula to rewrite it as \(1 - 2\sin^2\theta\), so the equation transforms into \(1 - 2\sin^2\theta + 4\sin^2\theta - 2\sin^4\theta = 1\)

Step 6: Simplify

This simplifies to \(2\sin^2\theta - 2\sin^4\theta = 0\)

Key Concepts

Double Angle Formula

The double angle formulas are a set of trigonometric identities used to express trigonometric functions of double angles in terms of single angles. These formulas can simplify complex trigonometric equations and are essential for solving a variety of calculus and geometry problems.

One of the key double angle formulas for cosine is: \[\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\] which can also be expressed in two other forms involving only sine or only cosine: \[\cos(2\theta) = 2\cos^2(\theta) - 1\] or \[\cos(2\theta) = 1 - 2\sin^2(\theta)\]
These identities become especially useful when dealing with trigonometric equations involving squared sine and cosine terms, as they allow for the conversion between different forms and can help to isolate variables for easier solving.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and variables, and the goal is usually to find all angles that satisfy the equation within a given interval. Solving these equations often requires the use of trigonometric identities, like the double angle formula, to rewrite the terms in a form that can be more easily manipulated.

When confronted with a trigonometric equation, it is common to follow a series of steps including: simplifying both sides, using identities to transform the equation, expanding and factoring expressions, and finally isolating the variable to find the solution. For example, in our original exercise, transforming \(\cos^2(2\theta)\) to \(1 - 2\sin^2(\theta)\) using the double angle formula is a crucial step in simplifying the equation and progressing towards a solution.

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle, and they are key to understanding relationships in triangles and modeling periodic phenomena. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), but there are also inverse functions, reciprocal functions, and hyperbolic functions associated with trigonometry.

In the context of our exercise, the cosine function describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle, and the sine function represents the ratio of the opposite side to the hypotenuse. Understanding these functions and their properties, such as their periodicity and symmetry, is vital for solving trigonometric equations and for real-life applications like signal processing and the study of wave patterns.