( tanx/1-cotx )+ (cotx/1-tanx)= (1+secxcscx)

Good one!

Generally these are done by changing everything to sines and cosines, unless you see some obvious identities.
Also generally, it is best to start with the more complicated side and try to change it to the "easier" side.

so here goes....

LS=
sinx/cosx ÷ (1-cosx/sinx) + cosx/sinx ÷(1-sinx/cosx)
=sinx/cosx [sinx/(sinx-cosx) + cosx/sinx [cosx/(cosx-sinx)
=sin^2 x/(cosx(sinx-cosx)) - cos^2 x/(sinx(sinx-cosx))
= (sin^3x - cos^3 x)/(sinxcosx(sinx-cosx))
=(sinx-cosx)(sin^2x + sinxcosx + cos^2)÷[sinxcosx(sinx-cosx)] difference of cubes!!!
= (sin^2x + cos^2x + sinxcosx)/(sinxcosx)
= (1 + sinxcosx)/sinxcosx

R.S.= 1+ 1/(sinxcosx)
=(sinxcosx + 1)/(sinxcosx) = L.S

Wow!!
=

thank you very much for your help .

You're welcome! I'm glad I could help you. The key to solving this equation is to simplify both sides and then show that they are equal. Let's go through the steps again in a more organized manner:

Starting with the left side (L.S.) and using the given expression:
L.S. = (tanx/1-cotx) + (cotx/1-tanx)

We can simplify this expression by replacing tangent and cotangent with sine and cosine, respectively:
L.S. = (sinx/cosx) / (1 - (cosx/sinx)) + (cosx/sinx) / (1 - (sinx/cosx))

Next, we can simplify the denominators by using the identity: 1 - (a/b) = (b - a)/b
L.S. = (sinx/cosx) / ((sinx - cosx)/sinx) + (cosx/sinx) / ((cosx - sinx)/cosx)

Now, we can simplify further by flipping the second fraction and multiplying:
L.S. = (sinx/cosx) * (sinx/(sinx - cosx)) + (cosx/sinx) * (cosx/(cosx - sinx))

Expanding the products gives us:
L.S. = (sin^2 x)/(cosx(sinx - cosx)) + (cos^2 x)/(sinx(cosx - sinx))

To continue simplifying, we can factor out common factors in the denominators:
L.S. = (sin^2 x)/(cosx * sinx - cos^2 x) + (cos^2 x)/(sinx * cosx - sin^2 x)

Now, we use the difference of squares identity: a^2 - b^2 = (a + b)(a - b)
L.S. = (sin^2 x)/((cosx - sinx) * (cosx + sinx)) + (cos^2 x)/((sinx - cosx) * (sinx + cosx))

Notice that the numerator in the first fraction is equal to the numerator in the second fraction. We can rewrite the expression now:
L.S. = (sin^2 x + cos^2 x) / ((cosx - sinx) * (cosx + sinx))

Using the identity sin^2 x + cos^2 x = 1, we have:
L.S. = 1 / ((cosx - sinx) * (cosx + sinx))

To simplify the denominator further, we use the difference of squares again:
L.S. = 1 / (cos^2 x - sin^2 x)

Using the Pythagorean identity cos^2 x - sin^2 x = cos 2x, we have:
L.S. = 1 / (cos 2x)

Now, let's simplify the right side (R.S.) using the given expression:
R.S. = 1 + secx * cscx

Since secx = 1/cosx and cscx = 1/sinx, we have:
R.S. = 1 + (1/cosx) * (1/sinx)

Combining the terms gives us:
R.S. = 1 + 1/(sinx * cosx) = (sinx * cosx + 1)/(sinx * cosx)

Comparing the L.S. and R.S. expressions, we can see that they are equal:
L.S. = 1 / (cos 2x) = (sinx * cosx + 1)/(sinx * cosx) = R.S.

Therefore, we have proved that the given expression is equal to (1 + secx * cscx).