(sinx - cosx)(sinx + cosx) = 2sin^2x -1
I need some tips on trigonometric identities. Why shouldn't I just turn (sinx + cosx) into 1 and would it still have the same identity?
I'm a little confused. Is the right equal to 2sin to the power of 2x-1?
Very simple
LS = sin^2 x - cos^2 x
= sin^2x - (1-sin^2 x)
= 2sin^2 x - 1
= RS
When simplifying trigonometric expressions, it's important to be aware of the trigonometric identities, as they can help you simplify the expression further. In this case, you have the expression (sinx - cosx)(sinx + cosx) on the left-hand side and 2sin^2x - 1 on the right-hand side.
To simplify the left-hand side, you can use the identity (a - b)(a + b) = a^2 - b^2. In this case, let's consider a = sinx and b = cosx. Using this identity, we can simplify as follows:
(sin^2x - cos^2x)
Now, let's take a look at your suggestion of turning (sinx + cosx) into 1. If you were to replace (sinx + cosx) with 1, the overall expression would become (sinx - cosx)(1). However, this step is not valid because it changes the expression's meaning. If you replace (sinx + cosx) with 1, you essentially ignore the original expression's content and substitute it with a constant, resulting in an incorrect identity.
Therefore, it's important to manipulate trigonometric expressions using valid trigonometric identities. In this case, the correct simplification is (sin^2x - cos^2x), which can be further simplified using the identity sin^2x + cos^2x = 1.
When working with trigonometric identities, it is essential to understand why certain manipulations are valid. Let's analyze the given expression and understand why you shouldn't simply turn (sinx + cosx) into 1.
Starting with the expression (sinx - cosx)(sinx + cosx), we can apply the distributive property to multiply the terms:
(sin^2 x - cos^2 x)
Now, we can use the Pythagorean identity, which states that sin^2 x + cos^2 x = 1. We'll rearrange it to isolate either sin^2 x or cos^2 x:
cos^2 x = 1 - sin^2 x
We substitute this into the expression we obtained earlier:
(sin^2 x - (1 - sin^2 x))
Now simplify:
sin^2 x - 1 + sin^2 x
This simplifies further to:
2sin^2 x - 1
Now, let's consider what would happen if you turned (sinx + cosx) into 1. It would result in:
(sinx - cosx)(1)
Which would simplify to:
sinx - cosx
As you can see, this is not the same as 2sin^2 x - 1. So, simply turning (sinx + cosx) into 1 doesn't maintain the original identity.
In summary, when manipulating trigonometric identities, it is vital to apply valid algebraic steps and remember that each step must preserve the original equation.