Question

If sin θ + cos θ = √3 , then tan θ + cot θ = ​

Answer 1

Answer:

sinθ+cosθ=√3

=(sinθ+cosθ)2=3

=sin2θ+cos2θ+2sinθcosθ=3

⇒2sinθcosθ=2

⇒sinθcosθ=1

⇒sinθcosθ=sin2θ+cos2θ

⇒1=sinθcosθsin2θ+cos2θ

⇒tanθ+cotθ=1

Answer 2

Explanation :

Note : We have taken this symbol → '∅' as theta.

Given,

Sin ∅ + cos ∅ = √3

Calculation :

sin ∅ + cos ∅ = √3 (squaring on both sides)

( sin ∅ + cos ∅ )² = (√3 )²

sin²∅ + cos²∅ + 2sin ∅ cos ∅ = 3

1 + 2 sin ∅ cos ∅ = 3

2 sin ∅ cos ∅ = 3 - 1

2 sin ∅ cos ∅ = 2

therefore, Sin ∅ Cos ∅ = 1

L.H.S. = tan ∅ + cot ∅

$\frac{ \sinΘ }{ \cosΘ } + \frac{ \cosΘ }{ \sinΘ } = \frac{ \sin^{2}Θ + { \cos }^{2}Θ}{ \sinΘ \cosΘ} \\ \frac{1}{ \sinΘ \cosΘ} = \frac{1}{1} \: \: \: \: \: ( \sinΘ \cosΘ = 1)$

L.H.S = R.H.S = tan ∅ + cot ∅ = 1

hence proved.