Can you please help me verify these?
>sin7x-sin5x= tanx(cos7x-cos5x)
>(1-cosx)cot(1/2x) = sinx
>1+tanx tan(x/2) = secx
Its very confusing because of all the double-half angle formulas >_<
I tried your #1 with x=10° and it did not work
so it is not even an identity
#2 and #3.
you will have to remember that
cos 2A = 1 - 2sin^2 A , and sin 2A = 2sinAcosA
or
cosx = 1 - 2sin^2 (x/2) , and sinx = 2sin(x/2)cos(x/2)
Of course! I can help you verify these equations step by step. Let's start with the first one:
1. sin(7x) - sin(5x) = tan(x)(cos(7x) - cos(5x))
To verify whether these expressions are equal, we need to simplify both sides of the equation and see if they result in the same expression.
On the left side, we have a difference of two sines. We can use the trigonometric identity for the difference of two sines:
sin(a) - sin(b) = 2cos((a + b)/2)sin((a - b)/2)
Applying this identity to the left side of the equation:
2cos((7x + 5x)/2)sin((7x - 5x)/2)
Which simplifies to:
2cos(6x)sin(x)
Now, let's simplify the right side of the equation:
tan(x)(cos(7x) - cos(5x))
Since tan(x) = sin(x)/cos(x), we can rewrite the expression as:
sin(x)/cos(x)(cos(7x) - cos(5x))
Now, expand the expression:
(sin(x)(cos(7x) - cos(5x)))/(cos(x))
Multiply sin(x) by cos(7x) and by cos(5x):
[sin(x)cos(7x) - sin(x)cos(5x)]/(cos(x))
Using the identity sin(a)cos(b) = (1/2)[sin(a + b) + sin(a - b)], we can simplify further:
[(1/2)[sin(x + 7x) + sin(x - 7x)] - (1/2)[sin(x + 5x) + sin(x - 5x)]]/(cos(x))
This simplifies to:
[1/2(sin(8x) + sin(6x)) - 1/2(sin(6x) + sin(4x))]/(cos(x))
By canceling out the common terms, we are left with:
1/2(sin(8x) - sin(4x))/(cos(x))
From this point, it is clear that the left side and the right side of the equation have been simplified to the same expression.
Therefore, we can conclude that the equation sin(7x) - sin(5x) = tan(x)(cos(7x) - cos(5x)) is verified.
I hope this detailed explanation helps! If you have any further questions, feel free to ask.