Question

2.1
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21. If sin 0 = 3 then find the value of tan? 6 + cot? 0 ?​

Answer 1

Answer:

\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} .

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