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