Question

\(\sin ^{6} A+\cos ^{6} A\) is equal to : (a) \(1-3 \sin ^{2} A \cos ^{2} A\) (b) \(1-3 \sin A \cos A\) (c) \(1+3 \sin ^{2} A \cos ^{2} A\) (d) 1

Short answer

Question: Simplify the expression \(\sin^6 A + \cos^6 A\). Choose the correct option. a) \(\sin^4 A - \cos^4 A\) b) \(\sin^2 A + \cos^2 A\) c) \(1 + 3 \sin^{2} A \cos^{2} A\) d) \(\sin^4 A + \cos^4 A\) Answer: The simplified expression for \(\sin^6 A + \cos^6 A\) is \(1 + 3 \sin^{2} A \cos^{2} A\). Therefore, the correct answer is option (c).

Step-by-step solution

Write given expression

Write down the given expression: \(\sin^6 A + \cos^6 A\)

Apply the Pythagorean identity

Use the Pythagorean identity, \(\sin^2 A + \cos^2 A = 1\), to represent \(\cos^2 A\) in terms of \(\sin^2 A\): \(\cos^2 A = 1 - \sin^2 A\)

Substitute into the given expression

Substitute this expression for \(\cos^2 A\) into the given expression and simplify: \((\sin^2 A)^3 + (1 - \sin^2 A)^{3}\)

Expand and simplify

Expand the expression using the binomial theorem and simplify: \((\sin^2 A)^3 + (1 - 3\sin^2 A + 3(\sin^2 A)^2 - (\sin^2 A)^3)\) Now, combine like terms and simplify further: \((1 - 3\sin^2 A + 3(\sin^2 A)^2)\)

Check for a match in the options

The simplified expression matches option (c): \(1 + 3 \sin^{2} A \cos^{2} A\) Therefore, the correct answer is option (c).

Key concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \(A\), the square of the sine plus the square of the cosine of the angle always equals 1. This can be written as: \[\sin^2 A + \cos^2 A = 1\] This identity is based on the Pythagorean theorem from geometry and is very handy for simplifying trigonometric expressions. For example, if you know \(\sin^2 A\), you can easily find \(\cos^2 A\) by rearranging the identity to \(\cos^2 A = 1 - \sin^2 A\).

• It relates the basic trigonometric functions in a straightforward formula.
• Useful for simplifying expressions and solving equations.
• It serves as a building block for more complex trigonometric identities.

In the original exercise, the identity is used to rewrite the expression \(\cos^2 A\) in terms of \(\sin^2 A\), which is crucial for simplifying the given expression.

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra for expanding expressions that are raised to a power. It is particularly useful when dealing with expressions of the form \((x + y)^n\). According to the theorem, this expression can be expanded into a series of terms involving the coefficients known as binomial coefficients. A general form of this expansion is: \[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\] Here, \(\binom{n}{k}\) represents the binomial coefficients, which can be calculated using combinations from combinatorics. The theorem allows each term in the series to be calculated efficiently.

• Allows expansion of expressions raised to powers without manual multiplication.
• Simplifies complex algebraic manipulations.
• Forms the basis for probability expansions and many algebraic calculations.

In the exercise, the binomial theorem helps expand \((1 - \sin^2 A)^3\) to simplify the expression further, making complex operations manageable and straightforward.

Simplification of expressions

Simplifying expressions is an essential math skill, as it involves reducing complex mathematical expressions into a simpler or more concise form. By removing unnecessary parts or combining like terms, simplification can make calculations more manageable and results clearer. In trigonometry, simplification often relies on identities and previously proven relationships, such as the Pythagorean identity and algebraic expansion. The goal is to transform the expression into a form that is usually more interpretable or fits a desired format.

• Reduces the expression to its simplest form.
• Enhances understanding and reduces potential calculation errors.
• Helps in identifying equivalent expressions and solving equations.

In the context of the exercise, simplification involves combining terms from the binomial expansion and using trigonometric identities. The process revealed the solution matching option (c), highlighting how simplification can direct us towards the correct answer.