\frac{\cos 70^\circ}{\sin 20^\circ} + \cos 36^\circ \csc 54^\circ

\frac{\cos 70^\circ}{\sin 20^\circ} + \cos 36^\circ \csc 54^\circ

The Correct Answer and Explanation is:

The expression we are looking to simplify is:cos⁡70∘sin⁡20∘+cos⁡36∘csc⁡54∘\frac{\cos 70^\circ}{\sin 20^\circ} + \cos 36^\circ \csc 54^\circsin20∘cos70∘​+cos36∘csc54∘

Step-by-Step Simplification:

1. First part: cos⁡70∘sin⁡20∘\frac{\cos 70^\circ}{\sin 20^\circ}sin20∘cos70∘​ We know from trigonometric identities that: cos⁡70∘=sin⁡20∘\cos 70^\circ = \sin 20^\circcos70∘=sin20∘ Therefore, the first part of the expression simplifies to: sin⁡20∘sin⁡20∘=1\frac{\sin 20^\circ}{\sin 20^\circ} = 1sin20∘sin20∘​=1
2. Second part: cos⁡36∘csc⁡54∘\cos 36^\circ \csc 54^\circcos36∘csc54∘ We know that: csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}cscθ=sinθ1​ Hence, csc⁡54∘=1sin⁡54∘\csc 54^\circ = \frac{1}{\sin 54^\circ}csc54∘=sin54∘1​ Therefore, the second part becomes: cos⁡36∘×1sin⁡54∘\cos 36^\circ \times \frac{1}{\sin 54^\circ}cos36∘×sin54∘1​ We also know that: sin⁡54∘=cos⁡36∘\sin 54^\circ = \cos 36^\circsin54∘=cos36∘ Thus, the second part simplifies to: cos⁡36∘×1cos⁡36∘=1\cos 36^\circ \times \frac{1}{\cos 36^\circ} = 1cos36∘×cos36∘1​=1

Final Answer:

So, the entire expression simplifies as follows:1+1=21 + 1 = 21+1=2

Thus, the value of the given expression is 2\boxed{2}2​.

Explanation:

We used known trigonometric identities to simplify the terms. First, we used the identity cos⁡70∘=sin⁡20∘\cos 70^\circ = \sin 20^\circcos70∘=sin20∘ to cancel out terms in the first fraction. In the second part, the identity sin⁡54∘=cos⁡36∘\sin 54^\circ = \cos 36^\circsin54∘=cos36∘ allowed us to simplify the expression for cos⁡36∘csc⁡54∘\cos 36^\circ \csc 54^\circcos36∘csc54∘, leading to a value of 1. Finally, the sum of the two parts results in 2.