Trigonometric Expressions and Identities

The video tutorial explains how to simplify the expression cosine to the fourth X minus sine to the fourth X. It begins by factoring the expression using the difference of squares method, resulting in two binomial factors. The Pythagorean identity is then applied to simplify one of the factors to 1. The tutorial proceeds with a substitution for cosine squared X, using the identity cosine squared theta equals one minus sine squared theta, leading to the final simplified expression of 1 minus 2 sine squared X.

What is the initial expression given in the problem?

• sine to the fourth X plus cosine to the fourth X

• sine squared X plus cosine squared X

• cosine to the fourth X minus sine to the fourth X

• cosine squared X minus sine squared X

Which algebraic method is used to factor the given trigonometric expression?

• Sum of cubes

• Sum of squares

• Difference of cubes

• Difference of squares

What are the two binomial factors obtained after factoring the expression?

• cosine squared X plus sine squared X and cosine to the fourth X minus sine to the fourth X

• cosine to the fourth X plus sine to the fourth X and cosine to the fourth X minus sine to the fourth X

• cosine X plus sine X and cosine X minus sine X

• cosine squared X plus sine squared X and cosine squared X minus sine squared X

What identity is used to simplify cosine squared X plus sine squared X?

• Reciprocal identity

• Quotient identity

• Pythagorean identity

• Even-Odd identity

What does cosine squared X plus sine squared X simplify to?

• cosine squared X

• 1

• 0

• 2

What substitution is made for cosine squared X during simplification?

• sine squared X plus 1

• sine squared X minus 1

• 1 plus sine squared X

• 1 minus sine squared X

What is the purpose of performing a substitution in the simplification process?

• To increase the complexity of the expression

• To express the equation in terms of a single trigonometric function

• To eliminate all trigonometric functions

• To convert the expression into a polynomial

After substitution, what does the expression simplify to?

• 1 plus 2 sine squared X

• 2 plus sine squared X

• 2 minus sine squared X

• 1 minus 2 sine squared X

Which identity from the list matches the final simplified expression?

• 2 plus sine squared X

• 2 minus sine squared X

• 1 minus 2 sine squared X

• 1 plus 2 sine squared X

What is the final step in the simplification process?

• Factoring the expression again

• Selecting the correct identity from the list

• Adding more trigonometric functions

• Multiplying the expression by a constant