Question

Eliminate α :
cot α + cos α = x
cot α - cos α = y

Answer 1

hello users. ....
I m using @ instead of alpha

solution:-
we have the given equations are
cot @ + cos @ = x ...(1.)
and
cot @ - cos @ = y ...(2.)

we have to eliminate @ from the equations .

Now,
adding (1.) &(2.) ...
=> 2cot @ = (x +y)
=> cot @ = (x+y)/2

now,
subtracting (2.) from (1.)
we get...
=> 2 cos @ = (x - y)
=> cos @ = (x-y) /2

Here,
since we have no square relation between cot @ and cos @ to eliminate @ from equations.

Taking reciprocal of cot @ and cos @
we get ...
tan @ = 1/cot @ = 2/(x+y) ...(3.)

and
sec @ = 1/cos @ = 2/ (x-y) ...(4.)

we know that:
sec²x - tan²x = 1

Now,
squaring and subtracting (3.) from (4.)
we get...
sec²@ - tan²@ = [ {2/(x-y) }² - {2/(x+y)}² ]

=> 1 = 4/(x-y)² - 4/ (x+y)²

=> 1 = [ {4(x+y)² - 4(x-y)²} / (x-y)²(x+y)² ]

..taking LCM...

=> (x-y)²(x+y)² = {4(x+y)² - 4(x-y)²}

...cross multiplying ....

=> [(x-y)(x+y)]² = 4[ x² + y² + 2xy - x² -y² +2xy ]

=> (x²-y²)² = 4×4xy

=> (x²-y²)² = 16 xy

which is required @ eliminated equation..

⭐✡ hope it helps :) ✡⭐