Question

if sin0 +cos0 =√3 ,then prove that tan 0 + cot0 =1​

Answer 1

Given---> Sinθ + Cosθ = √3

To prove ---> Tanθ + Cotθ = 1

Proof ---> We know that,

Sinθ + Cosθ = √3

Squaring both sides, we get,

=> ( Sinθ + Cosθ )² = (√3 )²

We have an identity ,

( a + b )² = a² + b² + 2ab , applying it here we get

=> Sin²θ + Cos²θ + 2Sinθ Cosθ = 3

We have a formula ,

Sin²A + Cos²A = 1 , applying it here , we get,

=> 1 + 2 Sinθ Cosθ = 3

=> 2 sinθ Cosθ = 3 - 1

=> 2 Sinθ Cosθ = 2

=> Sinθ Cosθ = 1

Now we know that,

tanA = SinA / CosA , CotA = CosA / SinA

Now, LHS

= tanθ + Cotθ = ( Sinθ / Cosθ ) + ( Cosθ / Sinθ )

= ( Sin²θ + Cos²θ ) / Sinθ Cosθ

Putting Sin²θ + Cos²θ = 1

= 1 / Sinθ Cosθ

Putting SinθCosθ = 1 , in it , we get

= 1 / 1

= 1 = RHS