Math 1A/1B. Pre-Calculus: Verifying Trigonometric Identities - Double Angle Formulas

Let's look at verifying trigonometric identities that involve the double angle formulas. For example, let's verify this identity. Well we can start working with the left hand side here. This is equal to sine x plus cosine x times sine x plus cosine x. And now we can FOIL, so this is sine squared x plus sine x cosine x and then plus cosine x sine x and then plus cosine squared. Now sine x cosine x and cosine x sine x are the same, so we'll have two of them. That is, this is equal to sine squared x plus 2 sine x cosine x and then plus cosine squared x. Now let's group these two together, in other words this is equal to sine squared x plus cosine squared x and then plus 2 sine x cosine x. Now why did we do that? Because remember we have the following Pythagorean identity. Sine squared x plus cosine squared x is 1, which we can use here. That is, this is just 1. So this is equal to 1 plus 2 sine x times cosine x. Now remember the double angle formula for sine, namely that sine 2x is equal to 2 sine x cosine x. So let's use that here, which means that this is equal to 1 plus sine 2x, which is our right hand side and therefore we verified this identity. Okay let's see another one. Let's verify this identity. We're going to start working here on the right hand side. Now looking in the denominator here, there is another Pythagorean identity that will help us. Namely 1 plus tangent squared x is equal to secant squared x. So let's use that here. Namely this is equal to 1 minus tangent squared x divided by secant squared x. And now let's divide both terms in the numerator by secant squared x, which gives us 1 divided by secant squared x minus tangent squared x divided by secant squared x. And now let's convert everything to sine and cosine. In other words this is equal to 1 over secant x is cosine x, and so 1 over secant squared x is cosine squared x. And then minus, now tangent x is sine x over cosine x, so tangent squared x is sine squared x over cosine squared x. And secant squared x is 1 over cosine squared x. Now what does this simplify to? Remember when we divide fractions it's the same as multiplying the numerator by the reciprocal of the denominator. In other words this is sine squared x divided by cosine squared x and then times cosine squared x over 1. Now won't these cosine squared x's cancel? So all we're left with then is just sine squared x. That is this is equal to cosine squared x minus sine squared x. Now let's recall all the double angle formulas. Unlike sine and tangent, cosine has 3 different double angle formulas. But if we look at them, what we just got is equal to the right hand side of the first one. Which means that what we just got is equal to cosine of 2x. So this is equal to cosine 2x, which is our left hand side. And so we've verified this identity. And this is how we verify trig identities that involve the double angle formulas. Thank you and we'll see you next time.