Without using trigonometric tables, evaluate the following:

$\text{cos}^2 \space 26° + \text{cos 64° sin 26°} + \dfrac{\text{tan 36°}}{\text{cot 54°}}.$

We need to find the value of

$\text{cos}^2\space26° + \text{cos 64° sin 26°} + \dfrac{\text{tan 36°}}{\text{cot 54°}}$

The above equation can be written as,

$\text{cos}^2\space26° + \text{cos (90 - 26)° sin 26°} + \dfrac{\text{tan 36°}}{\text{cot (90 - 36)°}}$

As, cos(90 - θ) = sin θ and cot(90 - θ) = tan θ. Using in above equation we get,

$\Rightarrow \text{cos}^2\space26° + \text{sin 26° sin 26°} + \dfrac{\text{tan 36°}}{\text{tan 36°}} \\[1em] = \text{cos}^2\space26° + \text{sin}^2\space26° + \dfrac{\text{tan 36°}}{\text{tan 36°}} \\[1em] = 1 + 1 \quad [\because \text{sin}^2\space A + \text{cos}^2\space A = 1] \\[1em] = 2.$

Hence, the value of the expression is 2.