Verify the equation is an identity using fundamental...

Verify the equation is an identity using fundamental identities and $\frac{A}{B} \pm \frac{C}{D}=\frac{A D \pm B C}{B D}$ to combine terms.
$$\frac{\cot x}{\sec x}-\frac{\cos x}{\sin x}=\frac{\cos x-1}{\tan x}$$

Key Concepts

Fundamental Trigonometric Identities

These are the core relationships between trigonometric functions, such as the definitions of sine, cosine, tangent, cotangent, secant, and cosecant. They allow the rewriting of complex trigonometric expressions in terms of primary functions, facilitating comparison or simplification in problems that require verification of identities.

Fraction Combination Using Common Denominators

This concept involves combining fractions by finding a common denominator, as expressed by the formula (A/B ± C/D = (A·D ± B·C)/(B·D)). This technique is essential for manipulating and simplifying expressions that involve sums or differences of fractional terms, especially when these terms contain trigonometric functions.

Algebraic Simplification

Algebraic simplification is the process of rewriting expressions in a more manageable or reduced form through techniques such as factoring, cancelation of like terms, and rearrangement. In verifying identities, simplifying each side of an equation using these strategies is critical to demonstrating that both sides are equivalent for all values in the domain.

Transcript

00:01 All right, in this problem, we're verifying this identity.

00:04 And normally it would start on the left side, but i think i'm going to do something just a little different.

00:07 I'm going to work with the right side.

00:10 So the right side and the left side are kind of equally complicated, but i feel like i can do maybe some interesting things with the right side to show you that, hey, i can work this either way.

00:22 So i'm going to break apart this fraction because i have tangent as the denominator.

00:28 I can break this apart like on subtract it or if you will and i'm left with cosine x over tan x minus 1 over tan x the cool thing about that second term is that's actually cotangent of x using the reciprocal identity for co -tendent in tangent the cosign x over tangent x i'm going to rewrite using a ratio or quotient identity and i have cosine x over sine x over cosine x over cosine x.

01:08 So i'm going to keep cosine x over one, change that to a division, and then flip.

01:23 All right.

01:24 And when i do that, i have, well, minus cotangent of x.

01:29 When i do that, i have a little bit of interesting things happening, right? i have cosine x over sine x, which i know is cotangent because i've done that in the first step.

01:45 But i'm trying to make it look like the left side, right? so what i'm going to do is do two things.

01:51 I'm going to change this thing in the box to cotangent x and then times cosine x minus.

02:00 Well, instead of writing cotangent x, i'm going to make the substitution of that ratio, right? up here in the blue up here now...

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