1

Problem

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

2

Step

The given expression

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

3

Step

First, simplify the given numerator:

\$1+cos(θ) - (1 - cos^2(θ))\$

i.e., \$(sin^2(θ) = 1 - cos^2(θ))\$

\$1+cos(θ) - 1+cos^2(θ)\$

\$cos(θ) + cos^2(θ)\$

4

Step

Now, simplify the denominator

sin(θ) + sin(θ)cos(θ)

5

Step

Now, plug these simplifications back into the original expression:

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

6

Step

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)))\$

7

Step

Now, we can cancel out the common term:

\$(cos(θ))/(sin(θ))\$

cot(θ)

8

Solution

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

9

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

10

Choice-A

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

Wrong \$- cot(\theta)\$

11

Choice-B

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

Wrong \$tan(\theta)\$

12

Choice-C

Option C is correct since it simplifies the cot⁡θ

Correct \$cot(\theta)\$

13

Choice-D

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

Wrong \$- tan(\theta)\$

14

Answer

Option

C

15

Sumup

Please summarize choices