Question

if sin theta + cos theta =root 3 then prove that tan theta + cot thera = 1

Answer 1

$\bold{\tan \theta+\cot \theta=1}$, If the value of  $\bold{\sin \theta+\cos \theta=\sqrt{3}}$

Given:

$\sin \theta+\cos \theta=\sqrt{3}$

To Prove:

$\tan \theta+\cot \theta=1$

Proof:

$\sin \theta+\cos \theta=\sqrt{3}$

Squaring of both sides, we get:

$(\sin \theta+\cos \theta)^{2}=(\sqrt{3})^{2}$

Using the formula $(a+b)^{2}=a^{2}+b^{2}+2 a b$

Applying formula in $(\sin \theta+\cos \theta)^{2},$

$\begin{array}{l}{\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=3} \\ {\because \sin ^{2} \theta+\cos ^{2} \theta=1}\end{array}$

1+2sinθcosθ=3

2sinθcosθ=2  

sinθcosθ=1  ______(1)

The value of the $\sin \theta+\cos \theta=\sqrt{3}$

is sinθcosθ=1  

To prove:

tanθ+cotθ=1  

L.H.S  

tanθ+cotθ

Transforming the identity of tanθ  ; cotθ into $\frac{\sin \theta}{\cos \theta}$ ; $\frac{\cos \theta}{\sin \theta}$

$\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}$

$\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta}$

Substituting equation (1) we get  

$\begin{array}{l}{\frac{\sin ^{2} \theta+\cos ^{2} \theta}{1}} \\ {\because \sin ^{2} \theta+\cos ^{2} \theta=1}\end{array}$

tanθ+cotθ=1=R.H.S

∴L.H.S=R.H.S

Hence proved

∴If $\bold{\sin \theta+\cos \theta=\sqrt{3}}$ then  $\bold{\tan \theta+\cot \theta=1}$ .

Answer 2

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