Without using trigonometric tables, prove that:

cosec² 67° - tan² 23° = 1

To prove,

cosec² 67° - tan² 23° = 1

Solving L.H.S. of the equation,

$\phantom{\Rightarrow} \text{cosec}^2 67° - \text{tan}^2 23° \\[1em] \Rightarrow \dfrac{1}{\text{sin}^2 67°} - \text{tan}^2 (90° - 67°) \\[1em] \Rightarrow \dfrac{1}{\text{sin}^2 67°} - \text{cot}^2 67° \\[1em] \Rightarrow \dfrac{1}{\text{sin}^2 67°} - \dfrac{\text{cos}^2 67°}{\text{sin}^2 67°} \\[1em] \Rightarrow \dfrac{\text{1 - cos}^2 67°}{\text{sin}^2 67°} \\[1em] \Rightarrow \dfrac{\text{sin}^2 67°}{\text{sin}^2 67°} \space [\because \text{ 1 - cos}^2 \text{ θ} = \text{sin}^2 \text{ θ}] \\[1em] \Rightarrow 1.$

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

Hence, proved that cosec² 67° - tan² 23° = 1.