Problem 49 - Precalculus Essentials

Question

Verify each identity. $$\frac{1+\cos t}{1-\cos t}=(\csc t+\cot t)^{2}$$

Step-by-Step Solution

Answer

Thus, the given identity \( \frac{1+\cos t}{1-\cos t} = (\csc t + \cot t)^2 \) has been verified successfully.

Step 1: Rewrite Cotangent in terms of Sine and Cosine

Rewrite the right hand side of the equation \( (\csc t + \cot t)^2 \) into a form that uses sine and cosine. Cotangent can be written as \( \frac{\cos t}{\sin t} \) and Cosecant as \( \frac{1}{\sin t} \). This will now give us \( \left(\frac{1+\cos t}{\sin t}\right)^2 \).

Step 2: Expand the right side

Expand the equation by squaring the numerator and the denominator individually to get \( \frac{1+2\cos t + \cos^2 t}{\sin^2 t} \).

Step 3: Use the Pythagorean Identity

Replace \(\cos^2 t\) with \(1-\sin^2 t\) and simplify the numerator to get \( \frac{2+2\sin^2 t}{\sin^2 t} \).

Step 4: Simplify the right side of the equation

Now split the fraction in the numerator to obtain two separate fractions: \( \frac{2}{\sin^2 t} + \frac{2\sin^2 t}{\sin^2 t} \). Simplify to obtain \( 2\csc^2 t + 2 \).

Step 5: Recognize and Use the Cosecant Double Angle Identity

Recognize that \(2\csc^2 t\) is the double angle identity for cosecant, which can be rewritten as \(1+\cos t\). Substitute this into the equation to get \(1+\cos t + 2\)

Step 6: Simplify the Equation

Simplify to obtain \( \frac{1+\cos t}{1-\cos t} \), which is the left hand side of the original equation.