Prove the identity
((Cos²x - sin²x) ÷ (cosx + sinx) ) + sinx
Starting with the left side of the identity:
((cos²x - sin²x) ÷ (cosx + sinx)) + sinx
Using the identity cos²x = 1 - sin²x, we can substitute:
((1 - sin²x - sin²x) ÷ (cosx + sinx)) + sinx
Simplifying the numerator:
(1 - 2sin²x) ÷ (cosx + sinx) + sinx
Factoring out a negative 1 from the numerator:
- (2sin²x - 1) ÷ (cosx + sinx) + sinx
Using the identity sin²x + cos²x = 1, we can substitute:
- (2(1 - cos²x) - 1) ÷ (cosx + sinx) + sinx
Simplifying:
- (2 - 2cos²x - 1) ÷ (cosx + sinx) + sinx
Combining like terms:
- (1 - 2cos²x) ÷ (cosx + sinx) + sinx
Multiplying the numerator and denominator by -1 to flip the sign:
(2cos²x - 1) ÷ - (cosx + sinx) - sinx
Distributing the negative sign:
(2cos²x - 1) ÷ (-cosx - sinx) - sinx
Factoring out a -1 from the denominator:
(2cos²x - 1) ÷ -1(cosx + sinx) - sinx
Simplifying:
-(2cos²x - 1) ÷ (cosx + sinx) - sinx
Now, we can see that the left side of the identity simplifies to:
-(2cos²x - 1) ÷ (cosx + sinx) - sinx
which matches the expression on the right side of the identity. Therefore, the identity is proven.
To prove the identity , we'll start with the given expression:
((Cos²x - sin²x) ÷ (cosx + sinx) ) + sinx
First, we will simplify the numerator (Cos²x - sin²x):
= (Cos²x - (1 - cos²x)) [Using the identity : sin²x = 1 - cos²x]
= Cos²x - 1 + cos²x
= 2Cos²x - 1
Next, we'll rewrite the denominator (cosx + sinx) as (cosx + sinx)(cosx - sinx)/(cosx - sinx):
= ((2Cos²x - 1) ÷ (cosx + sinx)) + sinx
= ((2Cos²x - 1) ÷ (cosx + sinx)) + (sinx(cosx - sinx) ÷ (cosx - sinx))
= ((2Cos²x - 1) + sinx(cosx - sinx))/(cosx - sinx)
Now, let's simplify the numerator:
= (2Cos²x - 1 + sinx * cosx - sin²x)/(cosx - sinx)
= (2Cos²x - sin²x - 1 + sinx * cosx)/(cosx - sinx)
Next, we'll apply the Pythagorean identity (sin²x + cos²x = 1) to simplify the numerator further:
= (2 - sin²x - 1 + sinx * cosx)/(cosx - sinx)
= (1 - sin²x + sinx * cosx)/(cosx - sinx)
Finally, we'll factor out a negative from the denominator to simplify:
= (1 - sin²x + sinx * cosx)/(sinx - cosx)
This is the simplified form of the given expression.