Question

If sin θ + cos θ = √3, then prove that tan θ + cot θ =1​

Answer 1

Step-by-step explanation

LHS=tan + cot

=sin/cos+cos/sin

=sin^2+cos^2/sin.cos

=1/sin.cos

we know,

cos=1/sin

so,

LHS=1

Hence,

LHS=RHS

Answer 2

Answer with Step-by-step explanation:

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

Taking LHS

$tan\theta+cot\theta$

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

By using formula

$tan\theta=\frac{sin\theta}{cos\theta},cot\theta=\frac{cos\theta}{sin\theta}$

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

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

$sin^2\theta+cos^2\theta+2sin\theta cos\theta=3$

By using $(a+b)^2=a^2+b^2+2ab$

$1+2sin\theta cos\theta=3$

By using identity $sin^2x+cos^2x=1$

$2sin\theta cos\theta=2$

$sin\theta cos\theta=\frac{2}{2}$

$sin\theta cos\theta=1$

Substitute the values

$\frac{1}{1}=1=LHS$

LHS=RHS

Hence, proved.

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