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.
Prove the identity
((Cos²x - sin²x) ÷ (cosx + sinx) ) + sinx
1 answer
1 answer