Prove that 1 + cot 60 degree upon 1 - Cos 60 degree ka whole square equal 1 + cos 30 degree upon 1 - cos 30 degree
To prove the given equation,
1 + cot 60° / (1 - cos 60°)^2 = 1 + cos 30° / (1 - cos 30°)
First, let's evaluate the values on both sides:
cot 60° = 1 / tan 60°
tan 60° = √3
cot 60° = 1 / √3
cot 60° = √3 / 3
cos 60° = 1/2
(1 - cos 60°)^2 = (1 - 1/2)^2 = (1/2)^2 = 1/4
cos 30° = √3/2
(1 - cos 30°) = (1 - √3/2) = (2 - √3) / 2
Now, substituting the values in the equation,
LHS = 1 + (√3 / 3) / (1 - 1/2)^2
= 1 + (√3 / 3) / (1/4)
= 1 + (√3 / 3) * (4/1)
= 1 + 4√3 / 3
RHS = 1 + (√3/2) / (2 - √3 / 2)^2
= 1 + (√3/2) / ((4 - √3) / 2)^2
= 1 + (√3/2) / ((4 - √3)^2 / 4)
= 1 + (√3/2) * (4/(4 - √3)^2)
= 1 + (4√3/2) * (4/(4 - √3)^2)
= 1 + 4√3 / 3
Since LHS = RHS, we have proven that:
1 + cot 60° / (1 - cos 60°)^2 = 1 + cos 30° / (1 - cos 30°)
To prove the given equation:
1 + cot(60°) / (1 - cos²(60°)) = 1 + cos(30°) / (1 - cos(30°))
Step 1: Simplify the left-hand side of the equation.
We know that cot(θ) = 1 / tan(θ), and tan(θ) = sin(θ)/cos(θ). Let's substitute these values into the equation:
1 + cot(60°) / (1 - cos²(60°))
= 1 + (1 / tan(60°)) / (1 - cos²(60°))
= 1 + (1 / (sin(60°) / cos(60°))) / (1 - cos²(60°))
Using the values of sin(60°) = √3/2 and cos(60°) = 1/2:
1 + (1 / (√3/2 / 1/2)) / (1 - (1/2)²)
= 1 + (1 / (√3 / 2)) / (1 - 1/4)
= 1 + 2/√3 / (3/4)
= 1 + 8/3√3
Step 2: Simplify the right-hand side of the equation.
Using the value of cos(30°) = √3/2:
1 + cos(30°) / (1 - cos(30°))
= 1 + (√3/2) / (1 - (√3/2))
= 1 + (√3/2) / (2/2 - √3/2)
= 1 + (√3/2) / (1/2)
= 1 + √3
= (√3 + 1) / 1
Step 3: Compare the left-hand side and right-hand side.
We have the left-hand side (LHS) = 1 + 8/3√3, and the right-hand side (RHS) = (√3 + 1) / 1.
Comparing LHS and RHS:
1 + 8/3√3 = (√3 + 1) / 1
To simplify the RHS, multiply the numerator and denominator by √3 - 1:
= (√3 + 1)(√3 - 1) / (1)(√3 - 1)
= (3 - 1) / (√3 - 1)
= 2 / (√3 - 1)
Since multiplying the numerator and denominator of the RHS by √3 - 1 gives us 2 / (√3 - 1), which is equal to LHS, we can conclude that:
1 + cot(60°) / (1 - cos²(60°)) = 1 + cos(30°) / (1 - cos(30°))