Without using trigonometric tables, evaluate the following:

$\dfrac{\text{sec 29°}}{\text{cosec 61°}} + \text{2 cot 8° cot 17° cot 45° cot 73° cot 82°} - \text{3(sin}^2 38° + \text{sin}^2 52°).$

We need to find the value of

$\dfrac{\text{sec 29°}}{\text{cosec 61°}} + \text{2 cot 8° cot 17° cot 45° cot 73° cot 82°} - \text{3(sin}^2 38° + \text{sin}^2 52°)$

The above equation can be written as,

$\dfrac{\text{sec 29°}}{\text{cosec (90 - 29)°}} + \text{2 cot 8° cot 17° cot 45° cot (90 - 17)° cot (90 - 8)°} - \text{3(sin}^2 (90 - 52)° + \text{sin}^2 52°)$

As, sin(90 - θ) = cos θ, cosec(90 - θ) = sec θ, cot(90 - θ) = tan θ, cot θ.tan θ = 1, cot 45° = 1 and sin²θ + cos²θ = 1. Using this in above equation we get,

$\Rightarrow \dfrac{\text{sec 29°}}{\text{sec 29°}} + \text{2 cot 8° cot 17° cot 45° tan 17° tan 8°} - \text{3(cos}^2 52° + \text{sin}^2 52°) \\[1em] = 1 + \text{2 cot 8° tan 8° cot 45° cot 17° tan 17°} - \text{3(cos}^2 52° + \text{sin}^2 52°) \\[1em] = 1 + 2 \times 1 \times 1 \times 1 - 3(1) \\[1em] = 1 + 2 - 3 \\[1em] = 3 - 3 \\[1em] = 0.$

Hence, the value of the above expression is 0.