Without using trigonometric tables, prove that:

sec² 22° - cot² 68° = 1

To prove,

sec² 22° - cot² 68° = 1

Solving, L.H.S. of the equation we get :

sec² 22° - cot² 68°

⇒ sec² 22° - cot² (90° - 22°)

We know that,

cot (90 - θ) = tan θ

⇒ sec² 22° - tan² 22°

$\Rightarrow \dfrac{1}{\text{cos}^2 22°} - \dfrac{\text{sin}^2 22°}{\text{cos}^2 22°} \\[1em] \Rightarrow \dfrac{\text{1 - sin}^2 22°}{\text{cos}^2 22°} \\[1em] \text{As, 1 - sin}^2 θ = \text{cos}^2 θ \\[1em] \Rightarrow \dfrac{\text{cos}^2 22°}{\text{cos}^2 22°} \\[1em] \Rightarrow 1.$

Since, L.H.S. = R.H.S.

Hence, proved that sec² 22° - cot² 68° = 1.