Trignometry questions?

2.

tanax = tanbx

=> ax = kπ + bx, k ∈ Z - {0}

=> x = kπ/(a - b), k ∈ Z - {0}

=> x = { ..... -2π(a - b), - π/(a - b), π/(a - b), 2π/(a - b), .....}

=> the solutions for a^2 + b^2 ≠ 0 are in A.P.

and common difference = π/(a - b)

3.

a cos 2θ + b sin 2θ = c

=> a (1 - tan^2 θ) / (1 + tan^2 θ) + 2b tan θ / (1 + tan^2 θ) = c

=> a (1 - tan^2 θ) + 2b tan θ = c (1 + tan^2 θ)

=> (c + a) tan^2 θ - 2b tan θ + (c - a) = 0

If α , β be the roots of the equation which is a quadratic in tan θ,

sum of the roots

= tan α + tan β = 2b/(c + a)

and product of the roots,

tan α * tan β = (c - a) / (c + a).

3. acos2x + bsin2x = c [ take theta = x ]

a + b tan2 x = c/cos2x

a + b [2tanx /1- tan^2x ] = c [1+tan^2x/1- tan^2x]

a [ 1- tan^2x] + b [2tanx] = c[ 1+ tan^2x]

a - atan^2x + 2b tanx = c + c tan^2x

tan^2x [a+c ] -- 2b tanx + {c-a} =0

now as tan alpha & tan beta are roots of this equation

so some of roots , i.e. tan alpha + tan beta = 2b/(c+a ) PROVED

and product of roots i. e. tan alpha * tan beta = c-a / c+a PROVED