Question

Write the formula to find angle between two straight lines y=m1x+c1 and y =m2x +c2

Answer 1

Given : angle between two lines y=m₁1x+c₁ and y=m₂x+c₂

To Find :derive an  expression for the angle between two lines

m₁ = Tanθ₁

m₂ = Tanθ₂

Tan (θ₂ - θ₁)  =(Tanθ₂ - Tanθ₁)/(1 +Tanθ₂Tanθ₁)

=> Tan (θ₂ - θ₁)  =  (m₂ - m₁)/(1 + m₂m₁)

=>  (θ₂ - θ₁) = Tan⁻¹{ (m₂ - m₁)/(1 + m₂m₁)}

angle between two lines y=m₁1x+c₁ and y=m₂x+c₂  is (θ₂ - θ₁) = Tan⁻¹{ (m₂ - m₁)/(1 + m₂m₁)}

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Answer 2

To Prove:

tanθ−secθ+1

tanθ+secθ−1

=

cosθ

1+sinθ

Solution:

L.H.S =

tanθ−secθ+1

tanθ+secθ−1

We can write, sec

2

θ−tan

2

θ=1

=

tanθ−secθ+1

tanθ+secθ−(sec

2

θ−tan

2

θ)

=

tanθ−secθ+1

tanθ+secθ−(secθ−tanθ)(secθ+tanθ)

=

tanθ−secθ+1

(tanθ+secθ){1−(secθ−tanθ)}

=

tanθ−secθ+1

(tanθ+secθ){1−secθ+tanθ}

=tanθ+secθ

=

cosθ

sinθ

+

cosθ

1

=

cosθ

1+sinθ

= R.H.S

since L.H.S = R.H.S

tanθ−secθ+1

tanθ+secθ−1

=

cosθ

1+sinθ

Hence Proved.