Question

if x = cot a + cos a and y= cot a - cot a . show that x^2 - y^2 = 4√xy.

Answer 1

Hi there!
Here's the answer:

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x = cot A + cos A
y = cot A - cos A

x - y = cot A + cos A - cot A + cos A
=> x - y = 2 cos A

x+y = cot A + cos A + cot A - cos A
=> x + y = 2 cot A

LHS = x² - y²
= (x+y)(x-y)
= (2cos A)(2 cot A)
= 4cos A cot A. --------(1)

RHS = 4√xy

xy= (cot A + cos A) (cot A - cos A)
= (cot² A - cos² A)
Take cos² A common and write cot² A = cos² A/sin² A
= cos² A[(1/sin² A) - 1]
= (cos²A/sin² A)[1 - sin² A]
= cot² A cos² A

√xy = cot A cos A
4√xy = 4cot A cos A -------(2)

From (1) & (2)

•°• LHS = RHS

Hence proved

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