Problem 1 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cos ^{4} A-\sin ^{4} A+1=2 \cos ^{2} A $$

Step-by-Step Solution

Answer

We can verify the trigonometric identity \(\cos^4 A - \sin^4 A + 1 = 2 \cos^2 A\) by using the fundamental trigonometric identity \(\sin^2 A + \cos^2 A = 1\). Solving for \(\sin^2 A\), we get \(\sin^2 A = 1 - \cos^2 A\). Substituting this into the given identity and simplifying, we obtain \(2\cos^2 A(\cos^2 A - 1) = 0\). This equation is true for any angle \(A\), thus verifying the given identity.

Step 1: Rewrite by using a trigonometric identity

The given identity is: \(\cos^4 A - \sin^4 A + 1 = 2 \cos^2 A\) To rewrite this identity, we will use the fundamental trigonometric identity: \(\sin^2 A + \cos^2 A = 1\) In this case, we can solve for \(\sin^2 A\) and substitute into the given identity: \(\sin^2 A = 1 - \cos^2 A\)

Step 2: Substitute to simplify the equation

Replace \(\sin^2 A\) with \(1 - \cos^2 A\) in the given identity: \(\cos^4 A - (1 - \cos^2 A)^2 + 1 = 2 \cos^2 A\)

Step 3: Further simplification and factoring

Now, let's expand and simplify the equation: \(\cos^4 A - (1 - 2\cos^2 A + \cos^4 A) + 1 = 2 \cos^2 A\) Combine like terms and subtract \(2 \cos^2 A\) from both sides: \(2\cos^4 A - 2\cos^2 A = 0\) Now, factor the equation: \(2\cos^2 A(\cos^2 A - 1) = 0\)

Step 4: Observe the equality

By inspecting the equation, we can see that when \(A\) is any angle, the left side of the equation is equal to the right side. Thus, the trigonometric identity is verified: \(\cos^4 A - \sin^4 A + 1 = 2 \cos^2 A\)

Key Concepts

Cosine and Sine Relationship

Understanding the relationship between cosine and sine is a cornerstone of trigonometry. These two functions are interdependent and connected through the fundamental Pythagorean identity, which states: \( \sin^2 A + \cos^2 A = 1 \). This identity essentially holds like a glue between the sine and cosine functions, manifesting their sum to be one in their squared forms.

For any angle \( A \), knowing either sine or cosine allows you to calculate the other. For instance, if you know \( \cos A \), then \( \sin A \) can be found using \( \sin A = \sqrt{1 - \cos^2 A} \). Remember, the square root function may yield both positive and negative values, depending on the quadrant where the angle resides.

This relationship is pivotal in transforming trigonometric identities. In our exercise, we used \( \sin^2 A = 1 - \cos^2 A \) to substitute \( \sin^2 A \) in the equation, which gracefully simplified the expression.

Polynomial Equations in Trigonometry

Trigonometric equations can often be polynomial-like in nature, meaning they involve powers and products of trigonometric functions. These equations require standard algebraic techniques to solve, such as simplifying or factoring.

In our example, the given equation was presented in the form \( \cos^4 A - \sin^4 A + 1 = 2 \cos^2 A \), which was simplified and manipulated to form a polynomial equation \( 2\cos^4 A - 2\cos^2 A = 0 \). This transformation highlights how powerful algebraic manipulation is when dealing with trigonometric equations.

By rewriting the terms, like substituting \( \sin^2 A \) with \( 1 - \cos^2 A \), we effectively convert the equation into a polynomial form, making it easier to identify gaps for factoring or further simplification. Mastering this process is essential for solving complex trigonometric equations.

Factoring Trigonometric Expressions

Factoring is a key strategy in solving polynomial equations, including those with trigonometric functions. It helps in reducing complex expressions into simpler, solvable parts. For our exercise, after simplifying the trigonometric equation, we reached \( 2\cos^2 A (\cos^2 A - 1) = 0 \).

The expression was factored to isolate potential solutions:

• \( 2\cos^2 A = 0 \), leading to \( \cos^2 A = 0 \)
• \( \cos^2 A - 1 = 0 \), leading to \( \cos^2 A = 1 \)

This factoring uncovers the possible values for \( \cos A \). The first factor implies that \( \cos A = 0 \), and the second gives the result that \( \cos A = \pm 1 \).

Factoring in trigonometry not only provides solutions but also validates identities by simplifying both sides of an equation to show that they are truly equal for all angles involved. It’s a vital tool in your trigonometric toolkit, ensuring you can break down and solve seemingly complex trigonometric expressions efficiently.