How do I prove that sinx/sinx+cosx=tanx/1+tanx
To prove that sinx / (sinx + cosx) = tanx / (1 + tanx), we can manipulate the expressions using trigonometric identities.
First, let's rewrite the right side of the equation using the identity tanx = sinx / cosx:
tanx / (1 + tanx) = (sinx / cosx) / (1 + sinx / cosx)
Now, let's find a common denominator for the fractions on the right side:
= (sinx / cosx) / (cosx / cosx + sinx / cosx)
= (sinx / cosx) / ((cosx + sinx) / cosx)
Next, let's invert the denominator and multiply:
= (sinx / cosx) * (cosx / (cosx + sinx))
= sinx / (cosx + sinx)
Finally, we can see that the right side simplifies to the left side:
sinx / (cosx + sinx) = sinx / (sinx + cosx)
Therefore, we have proved that sinx / (sinx + cosx) = tanx / (1 + tanx).
I am sure you meant to say
sinx/(sinx+cosx) = tanx/(1+tanx)
RS = (sinx/cosx)/[1 + sinx/cosx]
= (sinx/cosx)/[(cosx+sinx)/cosx]
= (sinx/cosx) ( cosx/(cosx + sinx))
= sinx/(cosx + sinx)
= LS