Verify that each equation is an identity.
csc A sin 2A -...

Question

Verify that each equation is an identity.

csc A sin 2A - sec A = cos 2A sec A

Accepted Answer

Hey, everyone here we are asked is the given equation and identity. We are given coin of X minus sine of two X multiplied by sin of X is equal to cosine of two X multiplied by coin of X. We have two answer choice options, answer a yes and answer B no. To determine if the equation is an identity, we need to verify that the left hand side is equivalent to the right hand side. So taking a closer look at our left hand side, we recall we have coin of X minus sine of two X multiplied by sin of X. So our goal is to manipulate this expression so that it, it matches the right hand side of the given equation. So the first thing we can notice is we have a sine of two X which tells us we need to recall the double angle formula for a sine function which tells us that sine of two X is equivalent to two sine of X multiplied by cosine of X. So now we can substitute in this expression into the left hand side of the equation. And so we have coin of X minus two multiplied by sine of X multiplied by cosine of X multiplied by sin of X. And now we want to rewrite sin in its other form. And so we need to recall that sin of X is equivalent to one divided by cosine of X. And so now just rewriting our current expression, we have cos of X minus two multiplied by sine of X multiplied by cosine of X multiplied by one divided by cosine of X. And now we can make some simplifications. So we can notice that we have cosine of X in the numerator and the denominator. So they cancel each other out. So, so far, we just have coin of X minus two sine of X. And so now we want to combine cos of X and two sin of X. So we want to first recall that cos of X is equivalent to one divided by sin of X. And so now rewriting we have one divided by sin of X minus two s of X. And now we can combine both of these terms by multiplying our two s of X by sin of X divided by sign of X. And with this, we have one divided by sine of X minus two sine squared of X divided by sine of X. And now we can see we have common denominators. So we can go ahead and combine the numerators. And so we have one minus two sine squared of X all divided by sine of X. And now we can notice that we can simplify our numerator by recalling the double angle formula for a cosine function which states that cosine of two X is equivalent to one minus two sine squared of X. So now just rewriting our expression, we have cosine of two X divided by sine of X. And now we just want to change back our one divided by sine of X into coin of X. And so we are left with here cosine of two X multiplied by coin of X. And with this, we can refer back to our original equation to its right hand side. And we can notice that our left hand side is in fact equivalent to the right hand side of the equation. And therefore, we can conclude that that yes, our given equation is in fact an identity. Therefore, we are left with answer choice. A where again, our answer is yes. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Verify that each equation is an identity.
csc A sin 2A - sec A = cos 2A sec A

Key Concepts

Trigonometric Identities

Reciprocal Identities

Double-Angle Formulas

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