Trigonometric Identities and Proofs Quiz

Which of the following is a trigonometric inverse identity?

• tan(θ) = sin(θ) / cos(θ)

• sin^2(θ) + cos^2(θ) = 1

• cosec(θ) = 1 / sin(θ)

• tan^2(θ) + 1 = sec^2(θ)

What is the result of dividing the Pythagorean identity by sin^2(θ)?

• tan^2(θ) + 1 = sec^2(θ)

• 1 + cot^2(θ) = cosec^2(θ)

• sin^2(θ) = 1 - cos^2(θ)

• cos^2(θ) = 1 - sin^2(θ)

In the first example, what is the first step in proving the identity?

• Use the Pythagorean identity

• Multiply both sides by cos(x)

• Separate the fraction into two parts

• Add 1 to both sides

What identity is used to simplify 1/sin(x) in the first example?

• tan(x) = sin(x) / cos(x)

• cosec(x) = 1 / sin(x)

• sec(x) = 1 / cos(x)

• cot(x) = cos(x) / sin(x)

In the second example, what is the expression for tan(θ) in terms of sine and cosine?

• tan(θ) = cos(θ) / sin(θ)

• tan(θ) = sin(θ) / cos(θ)

• tan(θ) = 1 / cos(θ)

• tan(θ) = 1 / sin(θ)

What is the result of multiplying the numerator and denominator by cos(θ) in the second example?

• sin^2(θ) + cos^2(θ)

• tan^2(θ) + 1

• 1 + cot^2(θ)

• sec^2(θ) - 1

In the third example, what is the first step in simplifying the left side of the equation?

• Use the Pythagorean identity

• Multiply by cos^2(θ)

• Factor out tan^2(θ)

• Add sec^2(θ) to both sides

What identity is used to simplify tan^2(θ) + 1 in the third example?

• tan^2(θ) = sec^2(θ) - 1

• tan^2(θ) + 1 = sec^2(θ)

• tan^2(θ) = 1 - sec^2(θ)

• tan^2(θ) = 1 + sec^2(θ)

What is the result of distributing sec^2(θ) in the third example?

• sec^4(θ) - sec^2(θ)

• sec^4(θ) + sec^2(θ)

• sec^2(θ) - 1

• sec^2(θ) + 1

What is the final expression on the right side of the equation in the third example?

• sec^2(θ) - 1

• sec^4(θ) + sec^2(θ)

• sec^4(θ) - sec^2(θ)

• sec^2(θ) + 1