Without using trigonometric tables, evaluate the following:

$\dfrac{\text{sin 35° cos 55° + cos 35° sin 55°}}{\text{cosec}^2 10° - \text{tan}^2 80°}$

We need to find the value of

$\dfrac{\text{sin 35° cos 55° + cos 35° sin 55°}}{\text{cosec}^2 10° - \text{tan}^2 80°}$

As, sin(90 - θ) = cos θ, cos(90 - θ) = sin θ, tan(90 - θ) = cot θ, sin²θ + cos²θ = 1 and cosec²θ - cot²θ = 1.

The above equation can be written as,

$\Rightarrow\dfrac{\text{sin 35° cos (90 - 35)° + cos 35° sin (90 - 35)°}}{\text{cosec}^2 10° - \text{tan}^2 (90 - 10)°} \\[1em] = \dfrac{\text{sin 35° sin 35° + cos 35° cos 35°}}{\text{cosec}^2 10° - \text{cot}^2 10°} \\[1em] = \dfrac{\text{sin}^2 35° + \text{cos}^2 35°}{\text{cosec}^2 10° - \text{cot}^2 10°} \\[1em] = \dfrac{1}{1} \\[1em] = 1.$

Hence, the value of the above expression is 1.