Simplify the expression using trig identities:

1. (sin4x - cos4x)/(sin2x -cos2x)

2. (sinx(cotx)+cosx)/(2cotx)

1.

I am sure you mean
(sin^4 x - cos^4 x)/(sin^2 x - cos^2 x)
= (sin^2 x + cos^2 x )((sin^2 x - cos^2 x)/(sin^2 x - cos^2 x)
= (sin^2 x + cos^2 x)
=1

2.
(sinx(cotx)+cosx)/(2cotx)
= (sinx(cosx/sinx) + cosx)/(2cosx/sinx)
= (cosx + cosx)(sinx/(2cosx)
= 2cosx(sinx)/(2cosx)
= sinx

To simplify the given expressions using trigonometric identities, we need to rewrite them in terms of basic trigonometric functions.

1. (sin4x - cos4x)/(sin2x - cos2x)

We can start by using the double angle formulas to rewrite sin4x and cos4x in terms of sin2x and cos2x:
sin4x = 2sin2xcos2x
cos4x = 2cos^2(2x) - 1 = 2(1 - sin^2(2x)) - 1 = 2 - 2sin^2(2x) - 1 = 1 - 2sin^2(2x)

Now, let's substitute these expressions back into the original expression:
[(2sin2xcos2x) - (1 - 2sin^2(2x))] / (sin2x - cos2x)

Expanding and rearranging the terms, we get:
(2sin2xcos2x - 1 + 2sin^2(2x)) / (sin2x - cos2x)

Next, we can factor out common terms from the numerator:
[(2sin2x - 1)(cos2x + 2sin2x)] / (sin2x - cos2x)

Now, we can simplify further by canceling out the common factor (sin2x - cos2x) from both the numerator and the denominator:
(2sin2x - 1)(cos2x + 2sin2x) / (sin2x - cos2x) = 2sin2x - 1

2. (sinx(cotx) + cosx) / (2cotx)

We can rewrite cotx as cosx/sinx, and substitute it back into the expression:
(sinxcosx/sinx + cosx) / (2cosx/sinx)

Simplifying, we get:
(cosx + cosx) / (2cosx/sinx) = 2cosx / (2cosx/sinx) = sinx

So, the simplified expression is sinx.

1. To simplify the expression (sin4x - cos4x)/(sin2x - cos2x), we can use the trigonometric identity sin^2(x) + cos^2(x) = 1.

First, let's rewrite the expression using this identity:
(sin^2(2x) - cos^2(2x))/(sin2x - cos2x)

Next, we can factor the numerator using the difference of squares formula:
((sin2x + cos2x)(sin2x - cos2x))/(sin2x - cos2x)

Now, we can cancel out the common factor of (sin2x - cos2x) in the numerator and denominator:
sin2x + cos2x

Therefore, the simplified expression is sin2x + cos2x.

2. To simplify the expression (sinx(cotx) + cosx)/(2cotx), we can use the trigonometric identities cotx = cosx/sinx and sin^2(x) + cos^2(x) = 1.

Let's start by substituting the value of cotx in terms of cosx and sinx:
(sin x * (cosx/sinx) + cosx)/(2 * (cosx/sinx))

Next, simplify the expression:
(cosx + cosx)/(2 * (cosx/sinx))
(2cosx)/(2 * (cosx/sinx))
(sin x * 2cosx)/(2cosx)
sin x

Therefore, the simplified expression is sinx.