Question

Which of the following expressions is not equivalent to cot^2θ?

a. (cosθ/sinθ)
b. sec^2θ - 1
c. csc^2θ - 1
d. 1/tan^2θ

Answer 1

Option c, csc^2θ - 1, is not equivalent to cot^2θ. The other options (a, b, d) are equivalent to cot^2θ due to definitions and Pythagorean identities in trigonometry.

Step-by-step explanation:

The expression that is not equivalent to cot^2θ among the given options is csc^2θ - 1. To verify this, let's examine each option:

• (cosθ/sinθ) is the definition of cotθ, and squaring it we get cot^2θ.
• sec^2θ - 1 is equivalent to cot^2θ due to the Pythagorean identities: sec^2θ is 1/cos^2θ, subtracting 1 gives us tan^2θ, and cotθ is 1/tanθ, so cot^2θ is indeed 1/tan^2θ.
• csc^2θ - 1 translates to 1/sin^2θ - 1 which is not equivalent to cot^2θ. The Pythagorean identity commonly associated with this expression is csc^2θ = 1 + cot^2θ.
• 1/tan^2θ is the reciprocal of tanθ, which defines cotθ, so squaring it yields cot^2θ.

Therefore, option c, csc^2θ - 1, is not equivalent to cot^2θ.