if cotx+tanx=a and secx-cosx=b then prove that (a^2b)^0.6667 -(ab^2)^0.6667=1
a^2b^(2/3) - (ab^2)^(2/3)
(a^2b)^(1/3) - (ab^2)^(1/3)(a^2b)^(1/3) + (ab^2)^(1/3)
ignoring the x's for a bit,
a^2 = (cot+tan)^2 = cot^2 + 2 + tan^2 = csc^2 + sec^2
a^2b = sec^3
b^2 = (sec-cos)^2 = sec^2 - 2 + cos^2
ab^2 = tan^3
(a2b)^(2/3) = sec^2
(ab^2)^(2/3) = tan^2
sec^2 - tan^2 = 1
To prove the given equation, we'll start by using trigonometric identities to represent cot(x), tan(x), sec(x), and cos(x) in terms of a and b:
Given:
cot(x) + tan(x) = a ----(1)
sec(x) - cos(x) = b ----(2)
Using the identity cot(x) = 1/tan(x), we can rewrite equation (1):
1/tan(x) + tan(x) = a
To simplify, let's use the substitution:
u = tan(x)
The equation then becomes:
1/u + u = a -----(3)
Next, we'll write sec(x) in terms of cos(x) using the identity sec(x) = 1/cos(x):
1/cos(x) - cos(x) = b
Using the substitution v = cos(x), the equation becomes:
1/v - v = b -----(4)
Now, let's solve equations (3) and (4) simultaneously to find u and v. To do this, we'll eliminate one variable by rearranging equation (3) to solve for 1/u as follows:
1/u = a - u
Substituting this into equation (4):
(a - u) - v = b
Rearranging this equation:
u + v = a - b
Now, let's square both sides:
(u + v)^2 = (a - b)^2
Expanding and simplifying:
u^2 + v^2 + 2uv = a^2 - 2ab + b^2
Since u = tan(x) and v = cos(x), we know that u^2 + v^2 = 1 (using the Pythagorean identity). So, we can rewrite the equation as:
1 + 2uv = a^2 - 2ab + b^2
Rearranging the terms:
2uv = a^2 - 2ab + b^2 - 1
Now, let's substitute back the original expressions for u and v:
2tan(x)cos(x) = a^2 - 2ab + b^2 - 1
Multiplying both sides by sin(x)*cos(x):
2sin(x)*cos(x)*tan(x) = (a^2 - 2ab + b^2 - 1)*sin(x)*cos(x)
Using the identity sin(2x) = 2sin(x)*cos(x):
sin(2x) = (a^2 - 2ab + b^2 - 1)*sin(x)*cos(x)
Multiplying both sides by 2:
2sin(2x) = 2(a^2 - 2ab + b^2 - 1)*sin(x)*cos(x)
Using the identity sin(2x) = 2sin(x)*cos(x):
4sin(x)*cos(x) = 2(a^2 - 2ab + b^2 - 1)*sin(x)*cos(x)
Dividing both sides by 2sin(x)*cos(x):
2 = a^2 - 2ab + b^2 - 1
Rearranging the terms:
a^2b - ab^2 = 1
Taking the cube root of both sides:
(a^2b - ab^2)^(2/3) = 1^(2/3)
(a^2b - ab^2)^(0.6667) = 1
Therefore, we have proved that (a^2b)^(0.6667) - (ab^2)^(0.6667) = 1