Question

If cot θ= 1; find the value of: 5 tan² θ+ 2 sin² θ- 3

Answer 1

Consider the diagram below :

cot θ = 1

i.e.`"base"/"perpendicular" = (1)/(1)`

Therefore if length of base = x, length of perpendicular = x

Since
base² + perpendicular² = hypotenuse² ...[ Using Pythagooras Theorem]

(x)² + (x)² = hypotenuse²

hypotenuse² = x² + x² = 2x

∴ hypotenuse =`sqrt2x`

Now

sin θ = `"perpendicular"/"hypotenuse" = (x)/(sqrt2x) = (1)/(sqrt2)`

tan θ = `"perpendicular"/"base" = (x)/(x) = 1`

Therefore
5tan² θ + 2sin² θ – 3

= `5 (1)^2 + 2 (1/sqrt2)^2 – 3`

= 5 + 1 – 3

=3