Question

if sin∅+cos∅=√3 then prove that tan∅+cot∅=1

Answer 1

Answer:

Question:

Given, Sinθ + Cosθ = √3

Prove that, Tanθ + Cotθ = 1

Solution:

Sinθ + Cosθ = √3

Now Do Squaring on both sides,

(Sinθ + Cosθ)² = (√3)²

Sin²θ + Cos²θ + 2SinθCosθ = 3

1 + 2SinθCosθ = 3

2SinθCosθ = 3 - 1

2SinθCosθ = 2

SinθCosθ = 2/2

SinθCosθ = 1.......(1)

Tanθ + Cotθ = 1

${ \sf{ \frac{sinθ}{cosθ } + \frac{cosθ}{sinθ} }}$

${ \sf{ \frac{ {sin}^{2} θ + {cos}^{2}θ }{sinθcosθ} }}$

${ \sf{ \frac{1}{sinθcosθ} }}$

From equation (1) :

${ \sf{ \frac{1}{1} = 1}}$

Hence proved

Answer 2

Question : -

Given, Sinθ + Cosθ = √3

Prove that, Tanθ + Cotθ = 1

Solution : -

Sinθ + Cosθ = √3

Now Do Squaring on both sides,

(Sinθ + Cosθ)² = (√3)²

Sin²θ + Cos²θ + 2SinθCosθ = 3

1 + 2SinθCosθ = 3

2SinθCosθ = 3 - 1

2SinθCosθ = 2

SinθCosθ = 2/2

SinθCosθ = 1... (1)

Tanθ + Cotθ = 1

$\begin{gathered}{ \sf{ \frac{sinθ}{cosθ } + \frac{cosθ}{sinθ} }} \\ \end{gathered} </p><p>cosθ</p><p>sinθ</p><p> </p><p> + </p><p>sinθ</p><p>cosθ$

$</p><p>\begin{gathered}{ \sf{ \frac{ {sin}^{2} θ + {cos}^{2}θ }{sinθcosθ} }} \\ \end{gathered} </p><p>sinθcosθ</p><p>sin </p><p>2</p><p> θ+cos </p><p>2</p><p> θ$

$</p><p>\begin{gathered}{ \sf{ \frac{1}{sinθcosθ} }} \\ \end{gathered} </p><p>sinθcosθ</p><p>1$

From equation (1) :

$\begin{gathered}{ \sf{ \frac{1}{1} = 1}} \\ \end{gathered} </p><p>1</p><p>1</p><p> </p><p> =1$

Hence proved.

Hope it will help you dear!!!

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