Trigonometric Identity

[Christine07]

I do not know the next step.There is no mistake in the numerator, the problem is stated directly from the textbook.Thank you for taking the time to assist me.

 

[Deleted member 4993]

View attachment 2670I do not know the next step.

I think there is a typo in your original problem . Check the exponents of the terms in the numerator - and repost.

 

[soroban]

Hello, Christine07!

\(\displaystyle \text{Prove: }\:\dfrac{\sec^4y - \tan^2y}{\sec^2y + \tan^2y} \:=\:1\)

There is no mistake in the numerator. . Really?
The problem is stated directly from the textbook.

As given, the equation is not true.
It must be a trick question.

If the numerator were \(\displaystyle \sec^4y - \tan^{\color{red}4}y\), we have an identity.

 

[DrPhil]

View attachment 2670I do not know the next step.There is no mistake in the numerator, the problem is stated directly from the textbook.Thank you for taking the time to assist me.

Looks like an error in the book itself.

It is easy enough to DISPROVE an identity - you just have to find one counterexample. For instance, try evaluating when y=30°.

As Soroban pointed out, it is an identity if the numerator has tan^4y. When I worked through that identity, I came down to cos^2y/cos^2y, which I happily canceled to 1. But I have a question for the purists (e.g. lookagain). What are we supposed to say about y=pi/2, or any value when cos^2y = 0? When we do trig identities, do we just gloss over such special cases, implicitly assuming that we are taking limits?

 

[mmm4444bot]

I have a question for the purists

What are we supposed to say about y=pi/2, or any value when cos^2y = 0? When we do trig identities, do we just gloss over such special cases

I'm not sure that this situation represents a special case. (The left-hand side of the original equation is not defined for some y values, like your Pi/2.)

I learned the definition of "an identity" as an equivalance that's true for all values of the variable for which both sides are defined.

Does this definition speak to your concern, or are you thinking of something else?

Cheers ~ Mark :cool:

 

[DrPhil]

I'm not sure that this situation represents a special case. (The left-hand side of the original equation is not defined for some y values, like your Pi/2.)

I learned the definition of "an identity" as an equivalance that's true for all values of the variable for which both sides are defined.

Does this definition speak to your concern, or are you thinking of something else?

Cheers ~ Mark :cool:

Thanks -- that is exactly the thought that hit me .. I was quite happy saying the identity was true for ALL y, but I have been corrected here a couple of times for forgetting the conditional "for which the function is defined."