Questions: 12:42 PM Fri Oct 18 a www-awu.aleks.com Midterm Exam (MATH114.001 - Fall 2024) Question 4 of 10 (1 point) Question Attempt: 1 of 1 Complete the proof of the identity by choosing the Rule that justifies each step. cot x(sec^2 x-1)=tan x To see a detailed description of a Rule, select the More Information Button to the right of the Statement Rule cot x(sec^2 x-1) = (cos x/sin x)(sec^2 x-1) Rule? Rule Algebra Reciprocal Quotient Pythagorean Odd/Even Continue

Solution Steps

To prove the identity \(\cot x(\sec^2 x - 1) = \tan x\), we can start by expressing \(\cot x\) and \(\sec^2 x\) in terms of sine and cosine. Then, simplify the expression using trigonometric identities and algebraic manipulation to show that both sides of the equation are equal.

We start with the left-hand side of the identity: \[ \cot x(\sec^2 x - 1) \] Using the definitions of \(\cot x\) and \(\sec^2 x\), we can rewrite this as: \[ \frac{\cos x}{\sin x} \left(\sec^2 x - 1\right) \]

Recall the Pythagorean identity: \[ \sec^2 x - 1 = \tan^2 x \] Thus, we can substitute this into our expression: \[ \cot x(\sec^2 x - 1) = \cot x \tan^2 x \]

Now substituting \(\tan x = \frac{\sin x}{\cos x}\) into the expression: \[ \cot x \tan^2 x = \frac{\cos x}{\sin x} \left(\frac{\sin^2 x}{\cos^2 x}\right) = \frac{\sin x}{\cos x} = \tan x \]

Final Answer