Can this Identity be verified by the Tangent Sum Identity?

[jddoxtator]

I am on a chapter where we are to verify equations are an identity with either the sum or difference identities.
Typically, the A or B value is a degree that does not make the equation undefined.
However, I have come across one that IS undefined.
I can see a path to prove it with the base Identity and a slight rearrangement of polarities, but I see no way to use the Sum Identity as instructed.

 

[R.M.]

Are you REQUIRED to prove it using sum/difference? I would use sine ([imath]\theta + \pi /2[/imath]) = cos[imath](\theta)[/imath] and cos([imath]\theta + \pi /2[/imath]) = -sin[imath](\theta)[/imath]

 

[jddoxtator]

Are you REQUIRED to prove it using sum/difference? I would use sine ([imath]\theta + \pi /2[/imath]) = cos[imath](\theta)[/imath] and cos([imath]\theta + \pi /2[/imath]) = -sin[imath](\theta)[/imath]

Now it does just say "Verify that each equation is an identity." Which is why I jumped to the base identity for Tan and Cot.
So I know there are other ways to prove it, just specifically this chapter is only dealing with the Sum and Difference identities and every other question is in terms of Sum and Difference identities.

 

[R.M.]

The two identities I cited can be easily shown with their respective sum identities, so it still "fits the theme" of that section.

 

[jddoxtator]

The two identities I cited can be easily shown with their respective sum identities, so it still "fits the theme" of that section.

So that would be just dealing with the numerator and denominator separately, correct?
What I did notice is that you can literally divide both sides of the original equation by negative one and get the base identity for tangent and cotangent which would also technically prove it, just not in terms that fit the theme.

 

[R.M.]

Yes, since tangent is a ratio of sin(x) to cos(x), I would deal with each separately. I would do the same with the difference identity.

 

[Dr.Peterson]

I am on a chapter where we are to verify equations are an identity with either the sum or difference identities.
Typically, the A or B value is a degree that does not make the equation undefined.
However, I have come across one that IS undefined.
I can see a path to prove it with the base Identity and a slight rearrangement of polarities, but I see no way to use the Sum Identity as instructed.

View attachment 38302

How about rewriting the LHS as [imath]\tan\left(\left(\theta-\frac{\pi}{2}\right)+\pi\right)[/imath]?

You can apply the angle-sum identity directly to this, along with the cofunction identity, and it produces the expected result.