Simplify #1:

cscx(sin^2x+cos^2xtanx)/sinx+cosx
= cscx((1)tanx)/sinx+cosx
= cscxtanx/sinx+cosx

Is the correct answer cscxtanx/sinx+cosx?

Simplify #2:
sin2x/1+cos2X
= ???

I'm stuck on this one. I don't know what I should do.

Simplify #3:
cosx-sin(90-x)sinx/cosx-cos(180-x)tanx
= cosx-(sin90cosx-cos90sinx)sinx/cosx-(cos180cosx+sinx180sinx)tanx
= cosx-sin90cosx+cos90sinxsinx/cosx-cos180cosx-sinx180sinxtanx
= cosx-sin90cosx+cos90sin^2x/cosx-cos180cosx-sinx180sinxtanx
= ???

What do I do next?

Please help and Thank you

#1

(1/sinx)(sin^2x + cos^2x(sinx/cosx) )/(sinx + cosx)
= (sinx + cosx)/(sinx+cosx)
= 1

#2
sin 2x/(1+ cos 2x)
= 2sinxcosx/(1 + 2sin^2x - 1)
= 2sinxcosx/2cos^2x
=sinx/cosx
= tan x

#3 new approach
remember that sin(90-x) = cosx
and cos(180-x) = -cosx , you are attempting to "prove" these in your solution

cosx-sin(90-x)sinx/cosx-cos(180-x)tanx
= cosx - cosx(tanx) - (-cosx)(tanx)
= cosx

I'm still confuse in #3

ok, let's go to your solution ....

from
cosx-(sin90cosx-cos90sinx)sinx/cosx-(cos180cosx+sinx180sinx)tanx
= cosx-((1)cosx-(0)sinx)(tanx) - ((-1)cosx+(0)sinx)tanx
= cosx - cosxtanx + cosxtanx
= cosx

I know where you're getting at but it's cosx-(sin90cosx-cos90sinx)sinx all over cosx-(cos180cosx+sin180sinx)tanx

sorry I should've put brackets to separate them.
[cosx-(sin90cosx-cos90sinx)sinx]/[cosx-(cos180cosx+sin180sinx)tanx]

To simplify an expression, we need to use the trigonometric identities and simplify each term step by step. Let's evaluate each expression you provided:

Simplify #1:
cscx(sin^2x + cos^2xtanx)/(sinx + cosx)
Using the identity sin^2x + cos^2x = 1,
cscx(1tanx)/(sinx + cosx)
The identity tanx = sinx/cosx,
cscx(sinx/cosx)/(sinx + cosx)
Multiplying the numerator and denominator by cosx to clear the fraction in the numerator,
cscxsinx/(sinx cosx) / (sinx + cosx)
Now, we can simplify the denominator (sinx cosx) with the identity sinx cosx = 1/2sin2x,
cscxsinx/(1/2sin2x) / (sinx + cosx)
Invert and multiply,
cscxsinx * (2sin2x) / (sinx + cosx)
Simplify the expression,
2sin^2x cscx / (sinx + cosx)
The reciprocal of cscx is sinx,
2sin^2x/sinx(cosx + sinx)
Finally, we can factor out sinx from the numerator,
2sinx(sinx)/(sinx(cosx + sinx))
The sinx in the numerator cancels out with the sinx in the denominator,
2sinx/(cosx + sinx)
Therefore, the simplified expression is 2sinx/(cosx + sinx).

Simplify #2:
sin2x/(1 + cos2x)
To simplify this expression, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (1- cos2x),
sin2x(1 - cos2x)/((1 + cos2x)(1 - cos2x))
Using the identity sin(2x) = 2sinx cosx and simplifying,
2sinx cosx(1 - cos2x)/(1 - cos^2(2x))
The identity cos^2(2x) = (1 - sin^2(2x)) can be employed,
2sinx cosx(1 - (1 - sin^2(2x)))/(1 - (1 - sin^2(2x))^2)
Simplifying further,
2sinx cosx(1 - 1 + sin^2(2x))/(1 - 1 + 2sin^2(2x) - sin^4(2x))
Canceling out terms,
2sinx cosx(sin^2(2x))/(2sin^2(2x) - sin^4(2x))
Factoring out sin^2(2x),
2sinx cosx(sin^2(2x))/(sin^2(2x)(2 - sin^2(2x)))
The sin^2(2x) in the numerator cancels out with the sin^2(2x) in the denominator,
2sinx cosx/(2 - sin^2(2x))
The identity sin^2(2x) = 1 - cos^2(2x) can be used,
2sinx cosx/(2 - (1 - cos^2(2x)))
Simplifying,
2sinx cosx/(1 + cos^2(2x))
Therefore, the simplified expression is 2sinx cosx/(1 + cos^2(2x)).

Simplify #3:
cosx - sin(90 - x)sinx/(cosx - cos(180 - x)tanx)
Let's evaluate this step by step.
Using the identity sin(90 - x) = cos(x),
cosx - cos(x)sinx/(cosx - cos(180 - x)tanx)
Next, we simplify cos(180 - x) using the identity cos(180 - x) = -cos(x),
cosx - cos(x)sinx/(cosx - (-cos(x))tanx)
cosx - cos(x)sinx/(cosx + cos(x)tanx)
Now, we apply the identity tanx = sinx/cosx,
cosx - cos(x)sinx/(cosx + cos(x)sinx/cosx)
Remembering to multiply the numerator and denominator by cosx to clear the fraction in the denominator,
cosx cosx - cos^2(x)sinx/(cosx cosx + cos(x)sinx)
cos^2(x) - cos^2(x)sinx/(cos^2(x) + cos(x)sinx)
The numerator cos^2(x) cancels out with cos^2(x)sinx,
- cos^2(x)sinx/(cos^2(x) + cos(x)sinx)
Finally, factoring out sinx,
- sinx cos^2(x)/(cos^2(x) + cos(x)sinx)
Therefore, the simplified expression is - sinx cos^2(x)/(cos^2(x) + cos(x)sinx).