Verify the identity.

cos 4u= cos^2(2u)- sin^2 (2u)

Question

Verify the identity.

cos 4u= cos^2(2u)- sin^2 (2u)

mathslover · Answer

Put 'u' as any angle ... let it be $\large{\frac{\pi}{2}}$ .

mathslover · Answer

What do you get now as : $\large{\cos(4(\frac{\pi}{2}))}$ = ?

mathslover · Answer

Can you tell me @mitchelsewbaran

.sam. · Answer

I would use 
cos(4u)
cos(2u+2u)
Then by identity: cos(A+B)=cosAcosB-sinAsinB
cos(2u)cos(2u)-sin(2u)sin(2u)
cos^2(2u)-sin^2(2u)

mathslover · Answer

Is that verification or proof @.Sam. I think we have to verify and not to prove

anonymous · Answer

cos(4(π 2  ))  = 1

.sam. · Answer

If they both equal then that's verifying too

mathslover · Answer

right now solve this : 
$$\large{\cos^2(2(\frac{\pi}{2}))- \sin^2(2(\frac{\pi}{2}))}$$

anonymous · Answer

hold on

mathslover · Answer

take your time

anonymous · Answer

1?

mathslover · Answer

yes can you tell me how you got that ?

anonymous · Answer

I replaced that whole equation with cos(2pi).

Taking the cosine of 2pi, I got 1

mathslover · Answer

o.O a small mistake @mitchelsewbaran :

see here:
$$\large{\cos^2(2(\frac{\pi}{2})) - \sin^2(2(\frac{\pi}{2}))}$$ 
$$\large{\cos^2(\pi) - \sin^2(\pi)}$$
$$\large{1-0= 1}$$

anonymous · Answer

oh srry about that

mathslover · Answer

No problem I just corrected you and I hope that you will not repeat that again.

Best of luck :)

anonymous · Answer

I have only 2 more that I need to verify. I was wondering if you can help?

tkhunny · Answer

You cannot substitute a single value (or any finite number of values) to PROVE this.  If you find one that doesn't work, that would be sufficient to DISprove it.

mathslover · Answer

Did you try yourself first?

anonymous · Answer

can u help me verify these last 2?

mathslover · Answer

there is not fight @mitchelsewbaran . sorry if you felt bad. 

@tkhunny the question is to verify .. and hence we have to put a value and verify it

anonymous · Answer

@mathslover ok :)

anonymous · Answer

no I didn't feel bad @mathslover

tkhunny · Answer

@mathslover  Demonstrating one or two values is just not what it is asking.  It must be verified for ALL values.  Since you cannot enter infinitely many values, there must be other methods employed.

Demonstrating $x = \pi$ says nothing of $x = \sqrt{2}$

mathslover · Answer

Ok @mitchelsewbaran post the one question of that two there in "ask a question forum" and surely I will help you but the second one will be solved by you and I will check it... if you get any problem in any concept you can ask me.

anonymous · Answer

ok :)

anonymous · Answer

@mathlover u're awesome, bro :)

mathslover · Answer

:)

anonymous · Answer

;)

mathslover · Answer

well I think @tkhunny has a point but still it is not given to verify for all values... it is just to verify.. but if we still go on for a proving method .. then it's surely not verifying

tkhunny · Answer

No, this is just not the case.

"Verify the Identity" has meant to provide a general proof for all possible values in the entire Domain - at least since 1972.  I am familiar with no reference in any mathematical text that has ever meant anything else.  If only a few values were to be sufficient, the problem statement would say something like this, "Verify a few values and formulate a conjecture on whether or not this statement is generally true."  "Verify the Identity" means ALL of them - leaving nothing out.