Problem 261 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \sin x+\sin ^{2} x=1, \text { show that } \cos ^{2} x+\cos ^{4} x=1 $$

Step-by-Step Solution

Answer

By using the Pythagorean trigonometric identity and some algebraic manipulations with the equations, one can arrive at the conclusion that if \(\sin x + \sin^2 x = 1\), then \(\cos^2 x + \cos^4 x = 1\).

Step 1: Manipulate the Given Equation

Rearrange the given equation to find \(\sin^2x\). The given equation is \(\sin x + \sin^2 x = 1\). By rearranging, we get: \(\sin^2 x = 1 - \sin x\)

Step 2: Substitution

Now substitute \(\sin^2 x = 1 - \sin x\) into the Pythagorean Trigonometric Identity, which is \(\sin^2 x + \cos^2x = 1\), we get: \(\cos^2 x = 1 - (1 - \sin x) = \sin x\)

Step 3: Analysing the Resultant Equation

Now, use the fact that \(\cos^{4}x + \sin^{4}x = \cos^{2}x\). Substitute \(\cos^2 x = \sin x\) into this equation to get: \(\cos^4 x + \sin^4 x = \sin x\)

Step 4: Final Simplification

The equation in step 3 simplifies to \(\cos^4 x + (1 - \cos^2x)^2 = \sin x\), which further simplifies to \(\cos^4 x + 1 - 2\cos^2x + \cos^4 x = \sin x\), and ultimately to \(\cos^2 x+\cos^4 x=1\)

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry between the sine and cosine functions. This identity is expressed as \( \sin^2 x + \cos^2 x = 1 \). It is named after the Pythagorean theorem because it similarly represents a fundamental relationship, in this case linking the squares of the sine and cosine of any angle. It's useful to break down the proof through this identity in any trigonometric problem.
In the given exercise, by using the rearranged equation \( \sin^2 x = 1 - \sin x \) from the original problem, we substitute into \( \sin^2 x \), allowing us to find \( \cos^2 x = \sin x \). This transformation plays a critical role in moving from one form to another and forms the basis for solving the equation step-by-step using algebraic manipulations. Understanding the Pythagorean identity gives you a solid ground to solve various trigonometric equations that are tied to a unit circle approach.

Sine and Cosine Relationships

Sine and cosine are core trigonometric functions representing angles based on the unit circle. These functions are interconnected through multiple identities and transformations that allow solving equations more flexibly.
In our exercise, the starting point was \( \sin x + \sin^2 x = 1 \); rearranging this gave \( \sin^2 x = 1 - \sin x \), which was pivotal for substitution into other identities. When expressed in terms of cosine, \( \cos^2 x \) becomes equal to \( \sin x \) given this specific setup. Such relationship manipulations reveal how sine and cosine can substitute each other within an equation's context due to this symmetry.

• Knowing how \( \sin x \) and \( \cos x \) can interchange helps simplify complex formulas into solvable steps.
• Recognizing \( \cos^2 x = \sin x \) helps transition into expressing powers of cosine terms as shown through the steps.

These transformations embody how deep sine and cosine intertwine, allowing shifts from sine-focused equations to those in cosine terms.

Trigonometric Equations

Trigonometric equations are used to find angles with specific trigonometric function values. Solving these equations often involves using identities to simplify conditions and express variables in terms of sine and cosine.
The given problem requires demonstrating that \( \cos^2 x + \cos^4 x = 1 \). After expressing \( \cos^2 x = \sin x \), we work through the deriving \( \cos^4 x + (1 - \cos^2 x)^2 = \sin x \).
A deeper look reveals the steps involve:

• Simplifying complex expressions into standard trigonometric identities.
• Utilizing relationships like \( \cos^4 x + \sin^4 x = \sin x \) to condense results.
• Reducing equations further to achieve the desired form of final expression \( \cos^2 x + \cos^4 x = 1 \).

This highlights the confluence of identities and relationships to simplify and resolve trigonometric structures effectively, a vital skill in understanding and solving trigonometric equations.