Explain how to determine the reduction identities from the double-angle identity cos(2 x)=cos^2

Explain how to determine the reduction identities from the...

Key Concepts

Double-Angle Identity

This concept involves formulas that express trigonometric functions of double angles in terms of the functions of single angles. In particular, the double-angle identity for cosine, cos(2x) = cos²x - sin²x, serves as a starting point to derive further identities by rearranging terms and applying other known trigonometric relationships.

Reduction Identities

Reduction identities are formulas that reduce expressions involving higher powers of trigonometric functions into ones involving functions of double angles or single angles. They are used to simplify integrals and solve equations by rewriting cos²x and sin²x in terms of cos(2x), thereby 'reducing' the power of the trigonometric expressions.

Pythagorean Identity

The Pythagorean identity, cos²x + sin²x = 1, is a fundamental relation in trigonometry. It is critical when deriving reduction identities because it allows one to express either sin²x or cos²x in terms of the other, facilitating the manipulation and rearrangement of the double-angle identity to solve for one of the squared terms.

Algebraic Manipulation

This key concept involves the process of rearranging equations, isolating variables, and substituting equivalent expressions to derive new identities. In the context of reduction identities, algebraic manipulation is applied to the double-angle identity and Pythagorean identity to express cos²x and sin²x solely in terms of cos(2x) and constants.

Transcript

00:01 Okay, in this problem, we are going to explain how to determine the reduction identities for the three trig function, sine, cosine, and tangent from the double angle identity that it gives us, which is the cosine of 2x equals cosine squared x minus sine x.

00:22 We can handle this.

00:23 We just need really a little bit of algebra, and we're going to use this formula that we have here for the pythagorean identity, sign squared x plus cosine squared x equals 1.

00:37 Here we go.

00:39 Let's start with the sign reduction identity.

00:42 To do that, we're going to solve for cosine in the pythagorean identity.

00:50 So we'll subtract sign from each side.

00:59 And this will give us cosine squared x equals 1 minus sine squared x.

01:08 We're going to simplify, we're going to substitute 1 minus sine squared x for cosine squared x in our double angle formula solve for sine squared x and that will give us our first reduction identity.

01:25 So here we go.

01:27 The cosine of 2x equals and we'll substitute 1 minus sine squared x minus sine squared x minus squared x and again all i did was substitute one minus sine for cosine right up there now we'll simplify we have like terms and sine squared x and sine squared x so cosine of 2x equals 1 minus 2 sine squared x now we'll just will use some algebra to solve for sine squared x first up is to subtract one from each side which will give us cosine of 2x minus 1.

02:25 It cancels equals negative 2 sine squared x.

02:31 Moving along with our algebra, we'll divide both sides by negative 2.

02:42 That cancels.

02:44 So we have cosine of 2x minus 1 divided by negative 2 equals sine.

02:54 Squared x and so now we have the sign squared isolated but we need to get the negative out of the denominator because it's just not convenient to have it down there so let's multiply the top and the bottom by negative one that doesn't change the value and i'll move my calculations over here to the right that will give me one minus the cosine of two 2x divided by 2 equals the sign squared of x.

03:32 And if you'll notice, we basically have the same thing.

03:35 We have up in black up top sign squared of x equals 1 minus a cosine of 2x divided by 2.

03:46 That's our first one.

03:48 For our second one, we're going to do the same thing, except we are going to solve for sign.

03:56 Substitute for sign.

03:57 So, and then solve for cosine squared.

04:00 So our first step is to subtract cosine squared from each side, which will leave us sine square of x equals 1 minus cosine square of x.

04:20 So we'll substitute that over in our double angle formula over here, the cosine of 2x equals the cosine squared of x and then we've got to be mindful of our negative minus parentheses 1 minus cosine squared x got to be careful not to drop that negative it'll mess you all up so our next step is to distribute the negative cosine of 2x equals cosine squared x minus 1 plus cosine squared x...

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