Product Identity: sin^2 t - tan^2 t = - (sin^2t)(tan^2t)

[baselramjet]

I'm a little lost on this one. My teacher explained it but I can't seem to figure it out.

in the equation:

sin^2 t - tan^2 t = - (sin^2t)(tan^2t)

is the product an identity or not an identity?

Any explanation would be greatly appreciated!

Ashley

 

[pka]

\(\displaystyle \L \begin{array}{rcl}
\left( {\sin ^2 (t)} \right)\left( {\tan ^2 (x)} \right) & = & \left( {1 - \cos ^2 (t)} \right)\left( {\tan ^2 (x)} \right) \\
& = & \left( {\tan ^2 (x) - \sin ^2 (x)} \right) \\
& = & - \left( {\sin ^2 (x) - \tan ^2 (x)} \right) \\
\end{array}\)

 

[baselramjet]

Thanks PKA, still a little lost, would that make the product an identity or not an identity?

 

[pka]

Thanks PKA, still a little lost, would that make the product an identity or not an identity?

I don't understand your point.
An identity is an identity is an identity.
Yes this is an identity.

 

[baselramjet]

I guess I'm asking about equality as far as functions are concerned, are the two functions equal...do they have the same value for every single value of t and how do you come about that?

 

[pka]

are the two functions equal...do they have the same value for every single value of t

What the word identity mean?
In what sense are two expressions identical?
Is this an identity: \(\displaystyle \sin ^2 (x) = 1 - \cos ^2 (x) ?\)

 

[baselramjet]

Right - in the equation:

sin^2 t - tan^2 t = - (sin^2t)(tan^2t) are both expressions identical?

is: sin^2 t - tan^2 t

the same as: - (sin^2t)(tan^2t)

and how do you arrive at the solution?

 

[pka]

how do you arrive at the solution?

\(\displaystyle \L \begin{array}{rcl}
- \left( {\sin ^2 (t)} \right)\left( {\tan ^2 (x)} \right) & = & - \left( {1 - \cos ^2 (t)} \right)\left( {\tan ^2 (x)} \right) \\
& = & - \left( {\tan ^2 (x) - \sin ^2 (x)} \right) \\
& = & \left( {\sin ^2 (x) - \tan ^2 (x)} \right) \\
\end{array}\)

 

[baselramjet]

does your solution mean they are not equal?

 

[Mrspi]

pka started wth the right-hand side, and by a legal substitution of 1 - cos^2 t for sin^2 t, and some algebraic manipulation, made it look exactly like the expression on the left-hand side. (I think that some of the t's inadvertently were typed as x's.....)

So, the two expressions say the same thing! That means setting them equal to each other results in a statement that is ALWAYS true for any values of t for which tan t is defined. Thus, it is an identity.

Perhaps you may wish to review the meaning of "identity."

 

[baselramjet]

Thank you for the explanation Mrspi! I was just unclear by the solution PKA presented without explanation.

The teacher was explaining this and was asking us to determine the identity of the product (hence - is the product an identity or not an identity)

And I got confused by how he was getting the answer for this particular equation.

Ashley

 

[stapel]

Thank you for the explanation Mrspi! I was just unclear by the solution PKA presented without explanation.

The complete worked solution was the complete solution. You were unclear because you didn't (apparently) know what an "identity" was.

This should (and normally would) have been covered extensively in class, and been defined clearly in your textbook. The tutor, naturally, had had no way of knowing that the term was never mentioned (or you would have asked for clarification in class), and was working under the perfectly-reasonable assumption that you'd heard the topic explained at some point.

Please forgive the confusion.

Eliz.