(tanA/1-cotA)+(cotA/1-tanA) = secAcosecA + 1 prove it

Sure, let's prove the given trigonometric identity step by step.

Given: (tanA/1-cotA)+(cotA/1-tanA) = secAcosecA + 1

Step 1: Convert all the trigonometric ratios to sine and cosine.

So, the given equation becomes:

(sinA/cosA)/(1-cosA/sinA) + (cosA/sinA)/(1-sinA/cosA) = 1/(sinAcosA) + 1

Step 2: Simplify the left-hand side (LHS) of the equation.

This gives us:

(sinA/cosA)(sinA/(sinA-cosA)) + (cosA/sinA)(cosA/(cosA-sinA))

Step 3: Further simplify the LHS.

This gives us:

(sin^2A/(cosA(sinA-cosA))) + (cos^2A/(sinA(cosA-sinA)))

Step 4: Take the LCM and simplify.

This gives us:

(sin^2A - cos^2A) / (cosA(sinA-cosA) - sinA(cosA-sinA))

Step 5: Use the identity sin^2A - cos^2A = -cos(2A) and simplify.

This gives us:

-cos(2A) / (cosA(sinA-cosA) - sinA(cosA-sinA))

Step 6: Simplify the denominator.

This gives us:

-cos(2A) / (cosAsinA - cos^2A - sin^2A + cosAsinA)

Step 7: Use the identity sin^2A + cos^2A = 1 and simplify.

This gives us:

-cos(2A) / (2cosA*sinA - 1)

Step 8: Use the identity 2cosA*sinA = sin(2A) and simplify.

This gives us:

-cos(2A) / (sin(2A) - 1)

Step 9: Multiply and divide by -1.

This gives us:

cos(2A) / (1 - sin(2A))

Step 10: Use the identity cos(2A) = 1 - 2sin^2A and sin(2A) = 2sinAcosA and simplify.

This gives us:

(1 - 2sin^2A) / (1 - 2sinAcosA)

Step 11: Simplify to get the right-hand side (RHS) of the equation.

This gives us:

1/(sinAcosA) + 1

Hence, the given equation is proved.