sorry the first one I put in had a typo
cosx/1-tanx + sinx/1-cotx = sinx +cosx
prove that it is an identity I assume ??
General rule to start:
1. begin with the complicated side and simplify it
2. change all trig ratios to sines and cosines if possible, unless an obvious relation can be seen
LS
= cosx/(1-sinx/cosx) + sinx/(1 - cosx/sinx)
= cosx/[(cosx - sinx)/cosx] + sinx/[(sinx-cosx)/sinx]
= cosx (cosx)/(cosx - sinx) - sinx (cosx - sinx)
= (cos^2 x - sin^2 x)/(cosx - sinx)
= (cosx - sinx)(cosx+ sinx)/(cosx - sinx)
= cosx + sinx
= RS
5th last line should say
= cosx (cosx)/(cosx - sinx) - sinx (sinx)(cosx - sinx)
Thanks!
No problem! Let's simplify the given expression:
cosx / (1 - tanx) + sinx / (1 - cotx) = sinx + cosx
First, let's simplify the left side of the equation. To do this, we need to rationalize the denominators.
Recall that tanx = sinx / cosx and cotx = cosx / sinx.
So, the expression becomes:
cosx / (1 - sinx / cosx) + sinx / (1 - cosx / sinx)
To simplify, we'll find the least common denominator (LCD) for the two fractions. In this case, the LCD is cosx * sinx.
For the first fraction, we multiply the numerator and denominator by cosx:
(cosx * cosx) / (cosx - sinx)
Similarly, for the second fraction, we multiply the numerator and denominator by sinx:
(sinx * sinx) / (sinx - cosx)
Now we can add the fractions:
[(cosx * cosx) / (cosx - sinx)] + [(sinx * sinx) / (sinx - cosx)]
To simplify this further, we need to combine the terms over a common denominator.
Multiply the numerator of the first fraction by (sinx - cosx) and the numerator of the second fraction by (cosx - sinx):
[(cosx * cosx) * (sinx - cosx) + (sinx * sinx) * (cosx - sinx)] / [(cosx - sinx) * (sinx - cosx)]
Expanding and simplifying the numerator:
[(cosx * cosx * sinx) - (cosx * cosx * cosx) + (sinx * sinx * cosx) - (sinx * sinx * sinx)] / [(cosx - sinx) * (sinx - cosx)]
Rearranging terms:
[(cosx * cosx * sinx) + (sinx * sinx * cosx) - (cosx * cosx * cosx) - (sinx * sinx * sinx)] / [(cosx - sinx) * (sinx - cosx)]
Now, factor out a negative from the last two terms of the numerator:
[(cosx * cosx * sinx) + (sinx * sinx * cosx) - (cosx * cosx * cosx) - (sinx * sinx * sinx)] / [(cosx - sinx) * (cosx - sinx)]
We can notice that the numerator is the same as the denominator:
[(cosx * cosx * sinx) + (sinx * sinx * cosx) - (cosx * cosx * cosx) - (sinx * sinx * sinx)] = [(cosx - sinx) * (cosx - sinx)]
Canceling out the common terms:
[(cosx - sinx) * (cosx - sinx)] / [(cosx - sinx) * (cosx - sinx)]
Now, the (cosx - sinx) terms in the numerator and denominator cancel out:
1
Therefore, the simplified expression is equal to 1:
cosx / (1 - tanx) + sinx / (1 - cotx) = 1