1. Prove the identity:cosx/1-tanx+sinx/1-cotx=sinx+cosx
2.Prove the identity: cosx/1+sinx+1+sinx/cosx=2secx
To prove an identity, we need to simplify one side of the equation until it matches the other side. Let's start by proving the first identity:
1. Proving the identity: cosx/(1 - tanx) + sinx/(1 - cotx) = sinx + cosx
To simplify the left side of the equation, we can first find the common denominator:
cosx/(1 - tanx) + sinx/(1 - cotx)
Now, let's find the LCD (Least Common Denominator), which in this case is (1 - tanx)(1 - cotx):
[(cosx * (1 - cotx)) + (sinx * (1 - tanx))] / [(1 - tanx)(1 - cotx)]
Expanding the numerator, we get:
[cosx - cosx * cotx + sinx - sinx * tanx] / [(1 - tanx)(1 - cotx)]
Now, let's simplify each term:
cosx * cotx = cosx * (1/tanx) = cosx/tanx = cosx * (cosx/sinx) = (cosx * cosx) / sinx = cos^2(x) / sinx
sinx * tanx = sinx * (1/cosx) = sinx/cosx = sinx * (sinx/cosx) = (sinx * sinx) / cosx = sin^2(x) / cosx
Substituting these simplified terms back into the expression:
[cosx - (cos^2(x) / sinx) + sinx - (sin^2(x) / cosx)] / [(1 - tanx)(1 - cotx)]
Now, let's simplify further:
Multiply through by sinx * cosx to get rid of the denominators:
[cosx * sinx - cos^2(x) + sin^2(x) - sinx * cosx] / [(1 - tanx)(1 - cotx)] * (sinx * cosx / sinx * cosx)
This becomes:
[sinx * cosx - cos^2(x) + sin^2(x) - sinx * cosx] / [(sinx * cosx)(1 - tanx)(1 - cotx)]
The terms sinx*cosx and -sinx*cosx cancel each other out, and sin^2(x) - cos^2(x) can be replaced using the identity sin^2(x) - cos^2(x) = sin(2x):
[0 + sin(2x)] / [(sinx * cosx)(1 - tanx)(1 - cotx)]
Simplify sin(2x) to get the final result:
sin(2x) / [(sinx * cosx)(1 - tanx)(1 - cotx)]
Since sin(2x) = 2 * sin(x) * cos(x), we can substitute it back into the expression:
[2 * sin(x) * cos(x)] / [(sinx * cosx)(1 - tanx)(1 - cotx)]
Now, we can simplify further:
2 / [(1 - tanx)(1 - cotx)]
Using the identity 1 - tanx = secx and 1 - cotx = cosecx:
2 / (secx * cosecx)
Multiply both the numerator and denominator by 2 to get rid of the fraction:
2 * secx * cosecx / 2
Finally, we are left with:
secx * cosecx
Since secx * cosecx = 1, we have proven the identity.
Therefore, the identity cosx/(1 - tanx) + sinx/(1 - cotx) = sinx + cosx holds true.