Proving y-z=a-c using tan x equation | Trigonometry Homework Help

[ritwik06]

Homework Statement

If $$tan x =\frac{2b}{a-c}$$
$$y=a.cos^{2} x + 2b.sin x.cos x+ c.sin^{2} x$$
$$z=a.sin^{2} x - 2b.sin x.cos x+ c.cos^{2} x$$Prove y-z=a-c

The Attempt at a Solution

Assuming that the result to be proved is true;

add y and z
y+z=a+c
and from the result to be proved
y-z=a-c
From this:
y=a
and
z=c

But can it be proved using the first equation of tan x ? I tried a lot but i couldn't do it.

 

[Dick]

Sure. Use the first equation to say 2b=(a-c)*tan(x). Substitute that for the 2b's in the other two equations.

 

[CompuChip]

First of all, you cannot assume what you want to prove. For example, I can prove 1 = 0 that way:

Assume that 1 = 0 (which we want to prove) is true. 
Then also 0 = 1. 
Add them: 1 = 1. 
This is a true equation, QED

.

Secondly, if you assume y-z=a-c, then why do you conclude y = a and z = c and even if this where true (which it's not) how would this help the proof?

You will have to work from the given information. First work out what y - z is. Then you can use the first equation of tan(x) to replace b by something in terms of a, c and x. Finally, use some more trig identities to get to a - c.

 

[ritwik06]

Sure. Use the first equation to say 2b=(a-c)*tan(x). Substitute that for the 2b's in the other two equations.

Thanks friend. It was silly of me to have not noticed that. Thanks a lot once again.