Answer:
See Below.
Step-by-step explanation:
We want to prove that:
[tex]\displaystyle \sin\theta\sec\theta+\cos\theta\csc\theta = 2\csc2\theta[/tex]
Let secθ = 1 / cosθ and cscθ = 1 / sinθ:
[tex]\displaystyle \frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}=2\csc2\theta[/tex]
Creat a common denominator. Multiply the first fraction by sinθ and the second by cosθ. Hence:
[tex]\displaystyle \frac{\sin^2\theta}{\sin\theta\cos\theta}+\frac{\cos^2\theta}{\sin\theta\cos\theta}=2\csc2\theta[/tex]
Combine fractions:
[tex]\displaystyle \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta}=2\csc2\theta[/tex]
Simplify. Recall that sin²θ + cos²θ = 1:
[tex]\displaystyle \frac{1}{\sin\theta\cos\theta}=2\csc2\theta[/tex]
Multiply the fraction by two:
[tex]\displaystyle \frac{2}{2\sin\theta\cos\theta}=2\csc2\theta[/tex]
Recall that sin2θ = 2sinθcosθ. Hence:
[tex]\displaystyle \frac{2}{\sin2\theta}=2\csc2\theta[/tex]
By definition:
[tex]\displaystyle 2\csc2\theta = 2\csc2\theta[/tex]
Hence proven.
