Prove cos7x+cos5x/sin7x-sin5x=cotx
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Gaurav Seth 5 years, 1 month ago
Left Hand Side =%20%2B%20cos(5x)%7D%7Bsin(7x)%20-%20sin(5x)%7D%7D%20%5C%5C)
%20%2B%20cos(t)%20%3D%202%20cos%20(%5Cfrac%7Bs%20%2B%20t%7D%7B2%7D)%20cos%20(%5Cfrac%7Bs%20-%20t%7D%7B2%7D)%7D%20%5C%5C)
%20-%20sin(t)%20%3D%202%20cos%20(%5Cfrac%7Bs%20%2B%20t%7D%7B2%7D)%20sin%20(%5Cfrac%7Bs%20-%20t%7D%7B2%7D)%7D%20%5C%5C)
%20cos%20(%5Cfrac%7B5x%20%2B%207x%7D%7B2%7D)%7D%7B2%20cos%20(%5Cfrac%7B5x%20%2B%207x%7D%7B2%7D)%20sin(%5Cfrac%7B-%205x%20%2B%207x%7D%7B2%7D)%7D%7D%20%5C%5C)
%20cos%20(%5Cfrac%7B12x%7D%7B2%7D)%7D%7B2%20cos%20(%5Cfrac%7B12x%7D%7B2%7D)%20sin%20(%5Cfrac%7B2x%7D%7B2%7D)%7D%7D%20%5C%5C)
therefore,
%20sin%20(%5Cfrac%7B12x%7D%7B2%7D)%7D%7Bcos%20(%5Cfrac%7B12x%7D%7B2%7D)%20sin%20(%5Cfrac%7B2x%7D%7B2x%7D)%7D%7D%20%5C%5C)
%7D%20%5C%5C)
and dividing the numbers: 
,
%7D%7Bsin%20(x)%7D%20%3D%20cot%20(x)%7D%20%5C%5C)
%7D%7D)
By applying the following identity of,
And, for the denominator's side, applying this identity,
Now,
Divide the following numbers
Now, cancelling out the common factor of:
Therefore, by the following trigonometric identity:
1Thank You