The value of cotθ.tan(90° - θ) - sec(90° - θ)cosecθ + (sin²25° + sin²65°) + $$\sqrt 3 $$ (tan5°. tan15°. tan30°. tan75°. tan85°)

Solution(By Examveda Team)

cotθ. tan(90° - θ) - sec(90° - θ)cosecθ + (sin²25° + sin²65°) + $$\sqrt 3 $$ (tan5°. tan15°. tan30°. tan75°. tan85°)
= cotθ. cotθ - cosecθ. cosecθ + (sin²25° + cos²25° ) + $$\sqrt 3 $$ [(tan5°. tan85°) . (tan15°. tan75°). tan30°]
  $$ = \left( {{\text{co}}{{\text{t}}^2}\theta - {\text{cose}}{{\text{c}}^2}\theta } \right) + \left( 1 \right) + \sqrt 3 \left( {1.1.\frac{1}{{\sqrt 3 }}} \right)$$         $$\left[ {{\text{tan A}}{\text{.tan B}} = {\text{1, If A}} + {\text{B}} = {{90}^ \circ }} \right]$$
$$\eqalign{ & = \left( { - 1} \right) + \left( 1 \right) + \sqrt 3 \times \frac{1}{{\sqrt 3 }} \cr & = - 1 + 1 + 1 \cr & = 1 \cr} $$