If tan A + cot A = 5; find the value of tan² A + cot² A.

tan A + cot A = 5

Squaring both sides,

(tan A + cot A)² = 5²

⇒ tan² A + cot² A + 2 x tan A x cot A = 25

⇒ tan² A + cot² A + 2 x tan A x $\dfrac{1}{\text{tan A}}$ = 25

⇒ tan² A + cot² A + 2 x ${\cancel{\text{tan A}}}$x $\dfrac{1}{\cancel{\text{tan A}}}$ = 25

⇒ tan² A + cot² A + 2 = 25

⇒ tan² A + cot² A = 25 - 2

⇒ tan² A + cot² A = 23

Hence, tan² A + cot² A = 23.