Problem 2 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ (\sin A+\cos A)(1-\sin A \cos A)=\sin ^{3} A+\cos ^{3} A $$

Step-by-Step Solution

Answer

To prove the equation \((\sin A+\cos A)(1-\sin A \cos A) = \sin^3 A+\cos^3 A\), we expand using the distributive property and use the trigonometric identity \(1 = \sin^2 A + \cos^2 A\). After simplifying and combining like terms, we end up with the expression \(\sin^3 A+\cos^3 A\) on both sides, thus proving the given equation to be true.

Step 1: Expand the Left Side of the Equation

To begin, we will expand the left side of the equation: \[ (\sin A+\cos A)(1-\sin A \cos A) \] Applying the distributive property, we get: \[ \sin A (1-\sin A \cos A) + \cos A (1-\sin A \cos A) \]

Step 2: Distribute Each Term

Now, we will distribute each term: \[ \sin A - \sin^2 A \cos A + \cos A - \sin A \cos^2 A \]

Step 3: Use Trigonometric Identity

We will apply the trigonometric identity \(1 = \sin^2 A + \cos^2 A\) by substituting \(\sin^2 A\) and \(\cos^2 A\) with the required expressions: We can replace \(\sin^2 A\) with \(1 - \cos^2 A\) and replace \(\cos^2 A\) with \(1 - \sin^2 A\): \[ \sin A - (1 - \cos^2 A) \cos A + \cos A - \sin A (1 - \sin^2 A) \]

Step 4: Distribute the Terms Again

Now, distribute the terms once more: \[ \sin A - \cos A + \cos^3 A + \cos A - \sin A + \sin^3 A \]

Step 5: Combine the Like Terms

Combine the like terms and simplify the expression: \[ \sin^3 A+\cos^3 A \] Now, the left side of the equation has been simplified to the same expression as on the right side of the equation. This shows that the two expressions are equal: \[ (\sin A+\cos A)(1-\sin A \cos A) = \sin^3 A+\cos^3 A \] Hence, we have proven the given equation to be true.

Key Concepts

Trigonometric Identity Proof

The process of proving a trigonometric identity involves showing that two different trigonometric expressions are equivalent regardless of the value of the angle involved. Proving such identities typically relies on a set of established trigonometric identities, such as the Pythagorean identities, reciprocal identities, quotient identities, and more. In our exercise example, the identity proof starts with expanding the left-hand side using the distributive property in trigonometry.
By methodically applying trigonometric identities and algebraic manipulation step by step, we simplify both sides of the equation to eventually match each other. In this particular exercise, after expanding the expression and simplifying, we end up with a cube identity expressed as \(\sin^3 A+\cos^3 A\), successfully proving the given identity. The key to such proofs is a strong understanding of basic identities and algebraic rules to combine and simplify terms effectively.

Distributive Property in Trigonometry

In trigonometry, the distributive property allows us to multiply a sum or difference of trigonometric functions by another function. This property is fundamental for expanding expressions, such as \(\sin A + \cos A\), when multiplied by \(1 - \sin A \cos A\) as in our original example. The property states that \(a(b + c) = ab + ac\), which we see applied in Step 1 of the solution.
Understanding the distributive property is crucial for simplifying trigonometric expressions, as it often serves as the initial step in trigonometric identity proof to ensure each term is correctly manipulated and combined. Misapplication of the distributive property is a common mistake, so it’s vital to apply it accurately to both trigonometric and non-trigonometric parts of an expression.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a recurring task in mathematics. It involves reducing expressions into a more manageable form by combining like terms, using basic arithmetic, and applying trigonometric identities. As exhibited in the provided solution, steps 3 and 4 focus on replacing squared trigonometric functions with their counterparts based on the identity \(1 = \sin^2 A + \cos^2 A\) and further simplifying.

• Recognize like terms to simplify expressions
• Apply appropriate trigonometric identities to make substitutions
• Be meticulous with signs and algebraic operations to avoid errors

In practice, these steps require careful attention to detail and a strong grasp of algebra and trigonometry. The final step involves combining like terms, at which point the expression should be in its simplest form, concluding the proof or simplification process.