Establish the identity. $$ (\csc \theta+\cot \theta)(\csc \theta-\cot \theta)=1 $$ Multiply and write the left side expression as the difference of two squares.

Question

UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ (\csc \theta+\cot \theta)(\csc \theta-\cot \theta) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\csc\left(\theta \right)+\cot\left(\theta \right)\right)\left(\csc\left(\theta \right)-\cot\left(\theta \right)\right)$$
- step1: Transform the expression:
  $$\frac{1+\cot\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \left(\csc\left(\theta \right)-\cot\left(\theta \right)\right)$$
- step2: Transform the expression:
  $$\frac{1+\cot\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}\times \frac{1-\cot\left(\theta \right)\sin\left(\theta \right)}{\sin\left(\theta \right)}$$
- step3: Multiply the terms:
  $$\frac{\left(1+\cot\left(\theta \right)\sin\left(\theta \right)\right)\left(1-\cot\left(\theta \right)\sin\left(\theta \right)\right)}{\sin\left(\theta \right)\sin\left(\theta \right)}$$
- step4: Multiply the terms:
  $$\frac{\left(1+\cot\left(\theta \right)\sin\left(\theta \right)\right)\left(1-\cot\left(\theta \right)\sin\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}$$
- step5: Transform the expression:
  $$\frac{\left(1+\cos\left(\theta \right)\right)\left(1-\cos\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}$$
- step6: Transform the expression:
  $$\frac{-\left(-\sin^{2}\left(\theta \right)\right)}{\sin^{2}\left(\theta \right)}$$
- step7: Calculate:
  $$\frac{\sin^{2}\left(\theta \right)}{\sin^{2}\left(\theta \right)}$$
- step8: Reduce the fraction:
  $$1$$
The identity $$(\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1$$ is established.

Now, let's simplify the left side expression by multiplying it:

$$(\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta$$

Therefore, the identity can be written as:

$$\csc^2 \theta - \cot^2 \theta = 1$$
$$ \csc^2 \theta - \cot^2 \theta = 1 $$

Establish the identity.

Multiply and write the left side expression as the difference of two squares.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Trigonometry Questions

Establish the identity. Multiply and write the left side expression as the difference of two squares.