We read your photo as:

Simplify the expression (1+cosθ)/sinθ * (cscθ - cotθ)

Mathematics

ScanSolve AI Answer

Answer

B. 1\text{B. } 1

Step-by-step solution

10 steps

To simplify the expression 1+cos⁡θsin⁡θ(csc⁡θ−cot⁡θ)\frac{1+\cos\theta}{\sin\theta}(\csc\theta - \cot\theta), we will follow these steps:

  1. Rewrite csc⁡θ\csc\theta and cot⁡θ\cot\theta in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta.

    Recall the definitions:

    csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta} cot⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta}

  2. Substitute these into the expression.

    1+cos⁡θsin⁡θ(1sin⁡θ−cos⁡θsin⁡θ)\frac{1+\cos\theta}{\sin\theta}\left(\frac{1}{\sin\theta} - \frac{\cos\theta}{\sin\theta}\right)

  3. Simplify the term inside the parenthesis by finding a common denominator.

    1sin⁡θ−cos⁡θsin⁡θ=1−cos⁡θsin⁡θ\frac{1}{\sin\theta} - \frac{\cos\theta}{\sin\theta} = \frac{1-\cos\theta}{\sin\theta}

  4. Substitute this simplified term back into the main expression.

    1+cos⁡θsin⁡θ(1−cos⁡θsin⁡θ)\frac{1+\cos\theta}{\sin\theta}\left(\frac{1-\cos\theta}{\sin\theta}\right)

  5. Multiply the numerators and the denominators.

    (1+cos⁡θ)(1−cos⁡θ)sin⁡θ⋅sin⁡θ\frac{(1+\cos\theta)(1-\cos\theta)}{\sin\theta \cdot \sin\theta}

  6. Simplify the numerator using the difference of squares formula, (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.

    Here, a=1a=1 and b=cos⁡θb=\cos\theta.

    (1+cos⁡θ)(1−cos⁡θ)=12−cos⁡2θ=1−cos⁡2θ(1+\cos\theta)(1-\cos\theta) = 1^2 - \cos^2\theta = 1 - \cos^2\theta

  7. Apply the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1.

    From this identity, we know that 1−cos⁡2θ=sin⁡2θ1 - \cos^2\theta = \sin^2\theta.
    So, the numerator becomes sin⁡2θ\sin^2\theta.

  8. Simplify the denominator.

    sin⁡θ⋅sin⁡θ=sin⁡2θ\sin\theta \cdot \sin\theta = \sin^2\theta

  9. Substitute the simplified numerator and denominator back into the expression.

    sin⁡2θsin⁡2θ\frac{\sin^2\theta}{\sin^2\theta}

  10. Simplify the fraction.

    Assuming sin⁡2θ≠0\sin^2\theta \neq 0, the expression simplifies to 1.

    The final answer is B. 1\text{B. } 1.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QSimplify the trigonometric expression: (sin 3θ + sin θ) / (cos 3θ + cos θ)QSimplify the trigonometric expression: ( (1 - cos(2x)) / sin(2x) ) * ( sin(2x) / (1 + cos(2x)) )QSimplify the expression (1 + cos 30°) / (1 - sin 60°).QSimplify the expression ( theta + 2theta)/(1 + theta + 2theta).
Browse by topic:AlgebraTrigonometry
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp