Which of the identities is true?

These are supposed to be IDENTITIES. An identity is true only if it is valid for every possible value of x. If there is even one value for x where the identity is invalid, it is false.

[math]sin \left ( \dfrac{\pi}{4} \right ) + cos \left ( \dfrac{\pi}{4} \right )= WHAT?[/math]
 
ugudansam said:
Using 1 for the last option. Sin1 + cos1 =1 that means sinx+ cosx = 1 is the correct answer?
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No !

Look at response #5 . Also, unless specified, trigonometric functions should be calculated using RADIANS.

Hint:

sin^2(x) = 1 - cos^2(x)

Which of the given identities can use the expression above?
 
ugudansam said:
View attachment 30622
Click to expand...
I would like to suggest a different approach.
1) [imath] \sin^2(x)=\sin(x)-1[/imath]
2) [imath]\sin(x)-sin(x)\cos^2(x)=sin^2(x)[/imath]
3) [imath]\sin(x)=\cos^2(x)+1[/imath]
4) [imath]\sin(x)+\cos(x)=1[/imath]
We know that ([imath]\forall x\in\Re)[-1\le\sin(x)\le 1][/imath] thus [imath]0\le\sin^2(x)\le 1[/imath] same for [imath]\cos^2(x)[/imath]
So in 1) the left side is non-negative but right side can be negative. think [imath]\left(x=\frac{5\pi}{4}\right)[/imath]
For 2) again [imath]\left(x=\frac{5\pi}{4}\right)[/imath] what is wrong there?
In 3) the right side is at least 1. why is that. what about the left side?
The graph of 4) is HERE. Is it ever negative? What wrong with that?


[imath][/imath][imath][/imath][imath][/imath][imath][/imath]
 
@pka you have have not copied the identity (2) correctly. It should have been:

sin(x) - sin(x) cos2(x) = sin3(x)
 
sin^2(x) > 0.

We know -1< sinx < 1
Subtract 1 from all sides---to get sinx - in the middle
-2 < sinx - 1 < 0

The only way sin^2(x) = sinx - 1 is when they are both 0. NOT for all x, so this is not an identity.
 
sin x = cos^2(x) + 1

-1< cosx < 1
So 0 < cos^2(x) < 1
1 < cos^2(x) 1+ < 2
So the right hand side of the possible identity is between 1 and 2 while sinx is between -1 and 1. Not an identity.
 
ugudansam said:
Actually that is exactly how be it was from the source
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Just to confirm:

sin(x) - sin(x) * cos2(x) = sin2(x) .........................................(1)

or

sin(x) - sin(x) * cos2(x) = sin3(x) .........................................(2)

Which one is correct

(exactly how be it was from the source) ​
​
- statement (1) or (2)?​
 
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