Verify that each equation is an identity (Hint: cos 2x = cos(x...

Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos( π/2 + x) = -sin x

Accepted Answer

Hello. Today we're going to be verifying whether the given equation is an identity. Now, in order to verify if an equation is identity, we need to prove that the left hand side of the equation is equal to the right hand side of the equation. Now let's take a look at the given equation. We are given cosine of three pi divided by two plus X is equal to sine of X. So how do we start this process? Well, if we take a look at the left hand side of the equation, we are given the sum property for cosine. So we can rewrite the left hand side of the equation using the sum property for cosine. Now recall that the sum property for cosine is given to us as cosine of A plus B is equal to cosine of a multiplied by cosine of B minus sine of A multiplied by sine of B. In order for us to use this property, we first need to identify the values of A and B. If we take a look at the function three pi divided by two is in the spot of A. So we're going to allow A to equal to three pi divided by two. Furthermore X sits in the spot of B. So we're going to allow B to equal to X now that we have the values of A and B, we can use the sum property for cosine and rewrite the left hand side of the equation as cosine of A which is three pi divided by two multiplied by cosine of B which is X minus sine of A, which is three pi divided by two multiplied by sine of B which is X and this is equal to sin of X. Now, we can further simplify the left hand side of the equation by evaluating cosine of three pi divided by two and sin of three pi divided by two. In order to do this, we're going to need the help of a unit circle. So in order to evaluate this, we first need to identify the angle of three pi divided by two on the unit circle. That angle will be located at the 270 degree mark of the unit circle. Furthermore, the point located at this angle is given to us as zero comma negative one. Now recall that cosine represents any X value of the terminal points on the unit circle and sine represents any Y value of the terminal points on the unit circle. What this means is that cosine of three pi divided by two is going to be the X value of the terminal point at the angle three pi divided by two, this value is going to be zero, which means that cosine of three pi divided by two will evaluate to give a zero sine of three pi divided by two will equal to the Y value of the terminal point at the angle three pi divided by two. This value is going to be negative one. So sin of three pi divided by two will evaluate to give us negative one. We can rewrite the left hand side of the equation as zero multiplied by cosine of X minus negative one multiplied by sine of X is equal to sign of X. Anything multiplied by zero will give us zero, negative one multiplied by negative one will give us positive one and positive one multiplied by sin of X will just give a sin of X. So the left hand side of the equation will simplify to sin of X which is equal to sin of X. This is a true statement meaning that this is an identity for cosine of three pi divided by two plus X. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to verify identities. And I'll go ahead and see you all in the next video.

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos( π/2 + x) = -sin x

Key Concepts

Trigonometric Identities

Angle Addition Formulas

Relationship Between Sine and Cosine

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