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Verify the following identities.$$\tan x+\cot x=2 \csc (2 x)$$

$\tan x+\cot x$ Verified

Precalculus

Algebra

Chapter 6

Trigonometric Identities, Inverses, and Equations

Section 4

The Double-Angle, Half-Angle, and Product-to-Sum Identities

Trigonometry

Functions

Missouri State University

McMaster University

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So this problem were asked to verify the given identity. And so when we verify identities, what we want is for the left hand side of the equation to exactly match the right hands out of the question. And so what? I mean by exactly matches. Let's say, for example, you know, Tangent of X is equal to sign X over coast I next, and so we'll sign x my next over Coastline X does in fact equal attention to vex. But to verify this, we need to somehow manipulate sign of eggs over coastline of X to get you get that two sides to exactly match in. So we rewrite sign of X over coastline of excess tangent of X. But we get tangent of X is equal to tangent of X. And so we verified. We have verified our problem, and so and so that's what we're going to do with this problem. So what we're gonna do is we're gonna work with the left hand side of the equation, try to make it look like the right hand side, and so so if we have tangent of X, remember, this is equal to sine X over coastline X and Co Tangent of X is equal to one over tangent of X, so it's going to be co sign of X over sine X. And so if we add these two together we get sign of X over coastline of X close co sign of X over a sign of X and so we can use the the butterfly Method. So the butterfly method is where we do b sine x times sign x first So we multiply Sign next time sine x and we're left with sine squared X science cortex and then we me ad We add co sign X Times coast under Vex But this goes coast on square next over And then we multiply any bottom so co sign x co sign X So I next sine x And so So this problem here Well, the the numerator in the numerator we have sine squared X plus coast on squared X And so by using the Pythagorean identity we know that this is one So sine squared explodes coastline scored X is equal to one and so there's co sign X sine x Well, if we know he and so working with the denominator now. And so if we know that that sign of two Alfa is equal to to sign Alfa Coastline Alfa, this is a double angle identity. Well, if we divide by two, we divide by two we get get that sign of two Alfa over two is equal to co sign about for time Sign of Alfa So this is what we have here We have a coastline x I neck So all I'm gonna do has changed the the outflow to become X So what we're left with If we put the numerator and denominator together, we get one over over Sign of two x over to this gives me this gives me two over signed two X and so So remember that Remember that that cosi can't co sequent of X is equal to equal to one over side of X. So So this means that co Sequent of two X is equal to equal to one over signed two x so to over signed two x go to over signed two x gives me too Cosi Kent of two x to co scenic to coast again of two X. And so now that we have that are left hand side and our right hand side match. We can verify this problems to co seek it of two x does in fact, equal to two coast. He can't of two x So this is our verification is this is the method to verify given identity.

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