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:
Identities:
cscx = 1/sinx
tanx = sinx/cosx
Your answer can be simplified further using the above identities.
#2:
This one looks simplified as is.
#3:
I'll let someone else help you with this one.
Simplify #2:
To simplify sin2x/1+cos2x, we can use the trigonometric identity sin^2x + cos^2x = 1. Rearranging this equation, we get 1 = 1 - cos^2x, which means cos^2x = 1 - sin^2x. Substituting this in the expression, we have:
sin2x / (1 + cos2x)
= sin2x / (1 + (1 - sin^2x))
= sin2x / (2 - sin^2x)
So the simplified expression is sin2x / (2 - sin^2x).
Simplify #3:
To simplify cosx - sin(90-x)sinx / cosx - cos(180-x)tanx, we can use the trigonometric identities:
sin(90 - x) = cosx
cos(180 - x) = -cosx
tanx = sinx / cosx
Substituting these identities, we have:
cosx - cosxsinx / cosx + cosx(sin^2x / cosx)
= cosx - cosxsinx / cosx + sin^2x
= (cosx - cosxsinx + sin^2x) / (cosx + sin^2x)
= (cosx(1-sinx) + sin^2x) / (cosx + sin^2x)
So the simplified expression is (cosx(1-sinx) + sin^2x) / (cosx + sin^2x).
For Simplify #1:
To simplify the expression cscx(sin^2x+cos^2xtanx)/sinx+cosx, we can start by simplifying the terms in the numerator and denominator.
The identity sin^2x + cos^2x = 1 is a basic trigonometric identity. Therefore, the expression can be simplified to cscx(1 + cos^2xtanx)/sinx+cosx.
Next, we apply the distributive property by multiplying cscx with both terms in the numerator. This gives us (cscx)(1) + (cscx)(cos^2xtanx)/sinx+cosx.
Since cscx is the reciprocal of sinx, we can rewrite the expression as 1/sinx + cscx(cos^2xtanx)/(sinx+cosx).
Finally, we can simplify cscx(cos^2xtanx) to cotx(cos^2xtanx) using the reciprocal trigonometric identity cscx = 1/sinx = cotx/cosx.
Therefore, the simplified expression becomes 1/sinx + cotx(cos^2xtanx)/(sinx+cosx).
For Simplify #2:
To simplify the expression sin2x/1+cos2x, we can start by recognizing that there is a trigonometric identity that involves the expression 1 + cos2x.
The identity 1 + cos2x = 2cos^2x states that the sum of the square of the cosine of an angle and 1 is equal to twice the square of the cosine of the same angle.
Therefore, we can rewrite the expression as sin2x/2cos^2x.
Next, we can use the identity sin2x = 2sinxcosx to further simplify the expression.
The expression now becomes (2sinxcosx)/(2cos^2x).
The 2 in the numerator and denominator cancel out, leaving us with sinxcosx/cos^2x.
Finally, we can simplify sinxcosx to 1/2sin2x using the identity sin2x = 2sinxcosx.
Therefore, the simplified expression is 1/2sin2x/cos^2x.
For Simplify #3:
To simplify the expression cosx-sin(90-x)sinx/cosx-cos(180-x)tanx, we can start by evaluating the trigonometric functions within the expression.
We know that sin(90 - x) is equal to cosx, and cos(180 - x) is equal to -cosx. tanx remains as it is.
So, the expression becomes cosx - cosxsinx/cosx + cosxtanx.
Next, we apply the distributive property to the numerator, giving us cosx - sinxcosx/cosx + cosxtanx.
We can factor out cosx in the numerator, which leaves us with cosx(1 - sinx)/cosx + cosxtanx.
Now, we can simplify the expression (1 - sinx)/cosx + cosxtanx.
To add fractions, we need a common denominator. In this case, the common denominator is cosx.
To get a common denominator, we can rewrite 1 - sinx as cosx - sinx.
Now, the expression becomes (cosx - sinx)/cosx + cosxtanx.
Combining the fractions, we have [(cosx - sinx) + cos^2x*tanx]/cosx.
Expanding further, we have [cosx - sinx + cos^2x*tanx]/cosx.
That is the simplified expression.