Prove :cosx-cosy= -2sin(x+y/2)sin(x-y/2)
Prove: 2cot-2tanx= 4-2sec^2x / tanx
2Sinx
To prove the given trigonometric identities, we will use basic trigonometric identities and algebraic manipulations. Let's start with the first identity:
Identity: cos(x) - cos(y) = -2sin((x+y)/2)sin((x-y)/2)
Proof:
We'll begin with the left side of the equation and try to manipulate it to match the right side of the equation.
Step 1: Expand cos(x) using the cosine difference formula.
cos(x) = cos((x+y)/2 + (x-y)/2) = cos((x+y)/2)cos((x-y)/2) - sin((x+y)/2)sin((x-y)/2)
Step 2: Expand cos(y) using the same formula.
cos(y) = cos((x+y)/2 - (x-y)/2) = cos((x+y)/2)cos((x-y)/2) + sin((x+y)/2)sin((x-y)/2)
Step 3: Subtract cos(y) from cos(x).
cos(x) - cos(y) = [cos((x+y)/2)cos((x-y)/2) - sin((x+y)/2)sin((x-y)/2)] - [cos((x+y)/2)cos((x-y)/2) + sin((x+y)/2)sin((x-y)/2)]
Step 4: Simplify the expression by cancelling terms.
cos(x) - cos(y) = cos((x+y)/2)cos((x-y)/2) - cos((x+y)/2)cos((x-y)/2) - [sin((x+y)/2)sin((x-y)/2) + sin((x+y)/2)sin((x-y)/2)]
Step 5: Combine like terms.
cos(x) - cos(y) = - 2sin((x+y)/2)sin((x-y)/2)
Therefore, the left side of the equation matches the right side of the equation, and we have proven the identity.
Now, let's move on to the second identity:
Identity: 2cot(x) - 2tan(x) = (4 - 2sec^2(x)) / tan(x)
Proof:
We'll start with the left side of the equation and manipulate it to match the right side.
Step 1: Expand cot(x) and tan(x) in terms of sine and cosine.
cot(x) = cos(x)/sin(x)
tan(x) = sin(x)/cos(x)
Step 2: Substitute the expanded forms into the equation.
2cot(x) - 2tan(x) = 2(cos(x)/sin(x)) - 2(sin(x)/cos(x))
Step 3: Find a common denominator for the fractions.
2cot(x) - 2tan(x) = (2cos(x)cos(x) - 2sin(x)sin(x)) / (sin(x)cos(x))
Step 4: Apply the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
2cot(x) - 2tan(x) = (2cos^2(x) - 2sin^2(x)) / (sin(x)cos(x))
Step 5: Factor out a common factor from the numerator.
2cot(x) - 2tan(x) = 2(cos^2(x) - sin^2(x)) / (sin(x)cos(x))
Step 6: Apply the difference of squares identity: a^2 - b^2 = (a + b)(a - b).
2cot(x) - 2tan(x) = 2((cos(x) + sin(x))(cos(x) - sin(x))) / (sin(x)cos(x))
Step 7: Cancel out the common factors.
2cot(x) - 2tan(x) = 2(cot(x) - tan(x))
Step 8: Simplify the expression.
2cot(x) - 2tan(x) = (4 - 2sec^2(x)) / tan(x)
Therefore, the left side of the equation matches the right side of the equation, and we have proven the identity.