How Do Trigonometric Identities Derive from Basic Formulas?

[aisha]

how would u proove this identity

$$\cos^2 \theta= \cos^2 - \sin^2= 2\cos^2\theta-1=1-2\sin^2\theta$$ ? The example i was given was $$\sin2\theta=2\sin\theta \cos\theta$$ this was a little bit easier to understand since there was a left and right side, I am not sure how to proove this. How was this formula derived from the sum differnce formula?

 

[vincentchan]

cos(a+b) = cos(a)cos(b)-sin(a)sin(b)
therefore
cos(2a)=cos(a)cos(a)-sin(a)sin(a)
this is the first step... the rest should be easy

 

[wais]

Proving identities in mathematics involves using different techniques such as algebraic manipulation, trigonometric identities, and substitution. In order to prove the given identity, we can use the Pythagorean identity which states that \sin^2\theta + \cos^2\theta = 1.

Starting with the left side of the given identity, we can use the Pythagorean identity to rewrite it as \cos^2\theta = 1 - \sin^2\theta. This is because if we subtract \sin^2\theta from both sides, we get \cos^2\theta + \sin^2\theta = 1 - \sin^2\theta + \sin^2\theta, which simplifies to \cos^2\theta = 1.

Next, we can use the difference of squares formula to rewrite \cos^2\theta as (\cos\theta + \sin\theta)(\cos\theta - \sin\theta). This gives us (\cos\theta + \sin\theta)(\cos\theta - \sin\theta) = 1 - \sin^2\theta.

Now, we can use the given identity \sin2\theta = 2\sin\theta\cos\theta to rewrite \sin^2\theta as \frac{1}{2}\sin2\theta. Substituting this into our previous equation, we get (\cos\theta + \sin\theta)(\cos\theta - \sin\theta) = 1 - \frac{1}{2}\sin2\theta.

To simplify the right side, we can use the double angle formula \sin2\theta = 2\sin\theta\cos\theta. This gives us (\cos\theta + \sin\theta)(\cos\theta - \sin\theta) = 1 - \sin^2\theta = 1 - 2\sin^2\theta.

Finally, we can rearrange the terms on the right side to get 1 - 2\sin^2\theta = 2\cos^2\theta - 1. This is the same as the second part of the given identity, proving that the two sides are equal.

In summary, we used the Pythagorean identity, difference of squares formula, and double angle formula to rewrite and simplify the left side of the given identity until it was equivalent to the right