Without using trigonometric tables, evaluate the following:

$\dfrac{\text{sin 65°}}{\text{cos 25°}} + \dfrac{\text{cos 32°}}{\text{sin 58°}} - \text{sin 28° sec 62°} + \text{cosec}^2 \space 30°$

We need to find the value of

$\dfrac{\text{sin 65°}}{\text{cos 25°}} + \dfrac{\text{cos 32°}}{\text{sin 58°}} - \text{sin 28° sec 62°} + \text{cosec}^2 \space 30°$

As sec θ = $\dfrac{1}{\text{cos θ}}$

The above equation can be written as,

$\dfrac{\text{sin 65°}}{\text{cos (90 - 65)°}} + \dfrac{\text{cos (90 - 58)°}}{\text{sin 58°}} - \text{sin 28°} \dfrac{1}{\text{cos (90 - 28)°}} + \text{cosec}^2 \space 30°$

As, cos(90 - θ) = sin θ and cosec 30° = 2. Using in above equation,

$= \dfrac{\text{sin 65°}}{\text{sin 65°}} + \dfrac{\text{sin 58°}}{\text{sin 58°}} - \text{sin 28°} \dfrac{1}{\text{sin 28°}} + 2^2 \\[1em] = 1 + 1 - 1 + 4 \\[1em] = 6 - 1 \\[1em] = 5.$

Hence, the value of the expression is 5.