Evaluate \$(1 + "cos"\theta - "sin"^2(\theta))/("sin"\theta(1 + "cos"\theta))\$.

The given expression

\$(1 + "cos"\theta - "sin"^2(\theta))/("sin"\theta(1 + "cos"\theta))\$

First, simplify the given numerator

⇒ \$1 + "cos"(θ) − (1 − "cos"^2(θ))\$
⇒ \$1 + "cos"(θ) − 1 + "cos"^2(θ)\$
⇒ \$"cos"(θ) + "cos"^2(θ)\$

Now, simplify the denominator

sin(θ) + sin(θ)cos(θ)

Now, plug these simplifications back into the original expression

\$("cos"(θ) + "cos"^2(θ))/("sin"(θ) + "sin"(θ)"cos"(θ))\$

Now, factor out a common term of cos⁡(θ) from the numerator and sin⁡(θ) from the denominator:

\$("cos"(θ)(1 + "cos"(θ)))/("sin"(θ)(1 + "cos"(θ)))\$

Now, we can cancel out the common term:

⇒ \$("cos"(θ))/("sin"(θ))\$

⇒ cot(θ)

Therefore, the value of the given expression is cot(θ).

Wrong: Because the expression evaluates to positive cot⁡(θ), not negative cot⁡(θ)

\$-"cot"(\theta)\$

This is wrong because the expression simplifies to cot⁡θ, not tanθ

\$"tan"(\theta)\$

Option C is correct since it simplifies the cot⁡θ

\$"cot"(\theta)\$

Option D is incorrect; simplifying the expression yields cotangent θ, not negative tangent θ

\$-"tan"(\theta)\$

Option

C

Can you summarize what you’ve understood in the above steps?