If (tan θ + cot θ) = 5, then (tan2 θ + cot2 θ) = ?

Question:

If (tan θ + cot θ) = 5, then (tan2 θ + cot2 θ) = ?

(a) 23
(b) 24
(c) 25
(d) 27

 

Solution:

(d) 23
We have (tan θ +cot θ) = 5
Squaring both sides, we get:
(tan θ +cot θ)2 = 52
⇒ tan2 θ + cot2 θ + 2 tan θ cot θ = 25

$\Rightarrow \tan ^{2} \theta+\cot ^{2} \theta+2=25 \quad\left[\because \tan \theta=\frac{1}{\cot \theta}\right]$

⇒ tan2 θ + cot2 θ = 25 − 2 = 23

 

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