Trigonometric Identities and Simplifications

The video tutorial explains how to simplify a trigonometric expression by selecting appropriate identities and performing substitutions. It begins with an expression involving cosine and sine, and through a series of substitutions and simplifications, it derives the final expression as cotangent x minus tangent x. The tutorial covers the use of double angle identities and demonstrates the process of simplifying complex fractions.

What is the initial expression given in the problem?

• 2 sine 2x divided by cosine 2x

• 2 cosine 2x divided by sine 2x

• 2 sine x divided by cosine x

• 2 tangent 2x divided by cotangent 2x

Which identity is used to substitute sine 2x?

• cosine 2θ = cosine² θ - sine² θ

• cotangent 2θ = (cotangent² θ - 1) / 2

• sine 2θ = 2 sine θ cosine θ

• tangent 2θ = 2 tangent θ / (1 - tangent² θ)

How many identities are available for substituting cosine 2x?

• Two

• Three

• Four

• One

Which identity is chosen for substituting cosine 2x in this problem?

• cosine 2θ = 1 + 2 sine² θ

• cosine 2θ = cosine² θ - sine² θ

• cosine 2θ = 2 cosine² θ - 1

• cosine 2θ = 1 - 2 sine² θ

What is the result of simplifying the expression into two fractions?

• sine x divided by sine x and cosine x divided by cosine x

• tangent x divided by cotangent x and cotangent x divided by tangent x

• cosine x divided by sine x and sine x divided by cosine x

• sine x divided by cosine x and cosine x divided by sine x

What does the first fraction simplify to?

• cosine x

• sine x

• cotangent x

• tangent x

What does the second fraction simplify to?

• tangent x

• cotangent x

• cosine x

• sine x

What is the final simplified expression?

• tangent x plus cotangent x

• cotangent x minus tangent x

• sine x minus cosine x

• cosine x plus sine x

Which choice does the final expression match?

• Second choice

• First choice

• Third choice

• Last choice

What is the purpose of performing substitutions in this problem?

• To find a matching expression among the given choices

• To convert the expression into a polynomial

• To make the expression more complex

• To eliminate all trigonometric functions