Show that .... is independent of theta

Bronn

Junior Member
Joined
Jan 13, 2017
Messages
62
Show that:


[(1 + cotø) / cscø] - [(1+tanø) / secø]


is independent of ø.


The above equals 0. But what does it mean to show that it is independent if ø? is it because no matter the value of ø the equation is always 0? help me understand how to think about this please!
 
Last edited:
I can't say as though I've seen the term "independent" used in this context before, but my best guess is that you're correct. If an expression can be simplified such that it no longer contains a particular variable, then it seems reasonable to say that expression is independent of the eliminated variable. Another example would be sin2(t) + cos^2(t) = 1, being that this is true for all values of t. Although, in this specific case, the expression doesn't seem to evaluate to 0 for every ø, as you say. For instance, plugging in ø = 1 shows that it is:

\(\displaystyle \left( 1 + \dfrac{cot(1)}{csc(1)} \right) - \left( 1 + \dfrac{tan(1)}{sec(1)} \right)\)

\(\displaystyle \approx \left( 1 + \dfrac{0.64}{1.19} \right) - \left( 1 + \dfrac{1.56}{1.85} \right)\)

\(\displaystyle \approx 1.54 - 1.84 \approx -0.3\)
 
Show that:


(1 + cotø / cscø) - (1+tanø / secø)


is independent of ø.


The above equals 0. But what does it mean to show that it is independent if ø? is it because no matter the value of ø the equation is always 0? help me understand how to think about this please!

I think the expression in your problem statement should be:

([1 + cotø] / cscø) - ([1+tanø ]/ secø) ...... those [] are super important for the problem to make sense.
 
Show that:


(1 + cotø / cscø) - (1+tanø / secø)


is independent of ø.


The above equals 0. But what does it mean to show that it is independent if ø? is it because no matter the value of ø the equation is always 0? help me understand how to think about this please!
(1 + cotø) / cscø - (1+tanø) / secø = 0?

Part 1:
(1 + cotø) / cscø
= (1+cosø/sinø) / (1/sinø)
= sinø + cosø

Part 2:
(1+tanø) / secø
= (1+sinø/cosø) / (1/cosø)
= cosø + sinø

Part 3:
Can you finish?
 
(1 + cotø) / cscø - (1+tanø) / secø = 0?

Part 1:
(1 + cotø) / cscø
= (1+cosø/sinø) / (1/sinø)
= sinø + cosø

Part 2:
(1+tanø) / secø
= (1+sinø/cosø) / (1/cosø)
= cosø + sinø

Part 3:
Can you finish?


(sinø + cosø) - (sinø + cosø) = 0



so if the equation equals 0 thats why its independent of ø?
 
Last edited:
I can't say as though I've seen the term "independent" used in this context before, but my best guess is that you're correct. If an expression can be simplified such that it no longer contains a particular variable, then it seems reasonable to say that expression is independent of the eliminated variable.
Right I see. So if the expression simplifies to answer like:

sin^2ø + cos^2ø = 1

then that expression is independent off ø because the result has dropped the ø?

cheers
 
Right I see. So if the expression simplifies to answer like:

sin^2ø + cos^2ø = 1

then that expression is independent off ø because the result has dropped the ø?

cheers
Correct
 
Top