MHB How Do You Solve These Trigonometric Identities?

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The discussion centers on solving the trigonometric identity (2 - 5cot x) / (2 + 5cot x) = (2sin x - 5cos x) / (2sin x + 5cos x). Participants question the accuracy of the identity and suggest that it should instead be written with cotangent on both sides. Two methods for solving the identity are proposed: multiplying the left side by sin x/sin x or the right side by csc x/csc x. Additionally, the relationships between cotangent, cosecant, sine, and cosine are highlighted as essential for simplifying the expression. The conversation emphasizes the importance of correctly identifying and manipulating trigonometric identities.
fluffertoes
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How do you do this one? I can't figure it out!
(2 - 5cot x) / (2 + 5cos x) = (2sin x - 5cos x) / (2sin x + 5cos x)
 
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Are you sure you've copied it correctly? It appears to me that the actual identity should be:

$$\frac{2-5\cot(x)}{2+5\cot(x)}=\frac{2\sin(x)-5\cos(x)}{2\sin(x)+5\cos(x)}$$
 
fluffertoes said:
How do you do this one? I can't figure it out!
(2 - 5cot x) / (2 + 5cot x) = (2sin x - 5cos x) / (2sin x + 5cos x)

fixedtwo ways to go

1) multiply left side by sinx/sinx

or

2) multiply right side by cscx/cscx

recall cotx = cosx/sinx and cscx = 1/sinx
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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