Prove the identity 1/cot²A + 1/(1 + tan²A) = 1/(1 − sin²A) − 1/csc²A.

Step-by-step solution

1. 1

   Choose the strategy: reduce both sides to sines and cosines separately. Neither side is obviously simpler than the other, so instead of transforming one into the other, simplify each independently and show the two results coincide. Every piece here is either a reciprocal identity or a Pythagorean identity.

2. 2

   Simplify the left-hand side. Since $\cot A=1/\tan A$, the first term is

   $\frac{1}{\cot^{2}A}=\tan^{2}A.$

   For the second, the Pythagorean identity $1+\tan^{2}A=\sec^{2}A$ gives

   $\frac{1}{1+\tan^{2}A}=\frac{1}{\sec^{2}A}=\cos^{2}A.$

   So the left-hand side is $\tan^{2}A+\cos^{2}A$.

3. 3

   Simplify the right-hand side. From $\sin^{2}A+\cos^{2}A=1$ we get $1-\sin^{2}A=\cos^{2}A$, so

   $\frac{1}{1-\sin^{2}A}=\frac{1}{\cos^{2}A}=\sec^{2}A,$

   and since $\csc A=1/\sin A$,

   $\frac{1}{\csc^{2}A}=\sin^{2}A.$

   So the right-hand side is $\sec^{2}A-\sin^{2}A$.

4. 4

   Bridge the two forms. Apply $\sec^{2}A=1+\tan^{2}A$ to the right-hand side:

   $\sec^{2}A-\sin^{2}A=1+\tan^{2}A-\sin^{2}A=\tan^{2}A+\bigl(1-\sin^{2}A\bigr)=\tan^{2}A+\cos^{2}A.$

   This matches the left-hand side exactly, so the identity holds.

5. 5

   State where it is valid, and test a value. The identity requires $\cot A$, $\tan A$ and $\csc A$ to be defined and $\cot A\ne0$, i.e. $A$ not a multiple of $\pi/2$. At $A=\pi/3$: the left side is $\tan^{2}=3$ plus $\cos^{2}=1/4$, giving $3.25$; the right side is $\sec^{2}=4$ minus $\sin^{2}=3/4$, giving $3.25$ ✓.