Answer:
See Below.
Step-by-step explanation:
11)
We have:
[tex]\displaystyle \cos\theta\cdot \cot\theta+\sin^2\theta\cdot \csc\theta=\csc\theta[/tex]
Rewrite:
[tex]\displaystyle \cos\theta\left(\frac{\cos\theta}{\sin\theta}\right)+\sin^2\theta\left(\frac{1}{\sin\theta}}\right)=\csc\theta[/tex]
Multiply:
[tex]\displaystyle \frac{\cos^2\theta}{\sin\theta}+\frac{\sin^2\theta}{\sin\theta}=\csc\theta[/tex]
Combine:
[tex]\displaystyle \frac{\cos^2\theta+\sin^2\theta}{\sin\theta}=\csc\theta[/tex]
According to the Pythagorean Identity, sin²θ + cos²θ = 1. Hence:
[tex]\displaystyle \frac{1}{\sin\theta}=\csc\theta[/tex]
Simplify:
[tex]\csc\theta=\csc\theta[/tex]
12)
We have:
[tex]\displaystyle \frac{\cos^2\theta +\sin^2\theta}{1+\tan^2\theta}=\cos^2\theta[/tex]
Pythagorean Identity:
[tex]\displaystyle \frac{1}{1+\tan^2\theta}=\cos^2\theta[/tex]
From the Pythagorean Identity, if we divide both sides by cos²θ, we acquire that tan²θ + 1 = sec²θ. Hence:
[tex]\displaystyle \frac{1}{\sec^2\theta}=\cos^2\theta[/tex]
Simplify:
[tex]\cos^2\theta =\cos^2\theta[/tex]
13)
We have:
[tex]\displaystyle \frac{\csc\theta\cdot \cos\theta}{\tan\theta +\cot\theta}=\cos^2\theta[/tex]
Rewrite:
[tex]\displaystyle \frac{1/\sin\theta\cdot \cos\theta}{\sin\theta/\cos\theta+\cos\theta/\sin\theta}=\cos^2\theta[/tex]
Multiply both top and bottom by cosθsinθ. Hence:
[tex]\displaystyle \frac{\cos^2\theta}{\sin^2\theta +\cos^2\theta}=\cos^2\theta[/tex]
By the Pythagorean Identity:
[tex]\displaystyle \frac{\cos^2\theta}{1}=\cos^2\theta[/tex]
Simplify:
[tex]\cos^2\theta=\cos^2\theta[/tex]
