Question

Cos α + cos β = a

sin α + sin β = b

show that

cos (α + β) = (a^2 - b^2)/(a^2 + b^2)

pls help me with this

and solve for x give in the picture
​

Answer 1

Explanation:

Let's start by evaluating a² − b² and a² + b².

a² − b²

= (cos α + cos β)² − (sin α + sin β)²

= cos² α + 2 cos α cos β + cos² β − (sin² α + 2 sin α sin β + sin² β)

= (cos² α − sin² α) + (cos² β − sin² β) + 2 (cos α cos β − sin α sin β)

= cos (2α) + cos (2β) + 2 cos (α + β)

a² + b²

= (cos α + cos β)² + (sin α + sin β)²

= cos² α + 2 cos α cos β + cos² β + sin² α + 2 sin α sin β + sin² β

= 2 + 2 (cos α cos β + sin α sin β)

= 2 + 2 cos (α − β)

cos (α + β)

Multiply top and bottom by 2 + 2 cos (α − β).

= cos (α + β) (2 + 2 cos (α − β)) / (2 + 2 cos (α − β))

= (2 cos (α + β) + 2 cos (α − β) cos (α + β)) / (2 + 2 cos (α − β))

Product to sum:

= (2 cos (α + β) + cos (α + β − (α − β)) + cos(α + β + α − β)) / (2 + 2 cos (α − β))

= (2 cos (α + β) + cos (2β) + cos (2α)) / (2 + 2 cos (α − β))

Substitute:

= (a² − b²) / (a² + b²)

atan(1/x) − atan(2 − x) = π/12

Angle difference formula:

atan((1/x − (2 − x)) / (1 + (1/x)(2 − x))) = π/12

atan((1/x − 2 + x) / (1 + 2/x − 1)) = π/12

atan((1/x − 2 + x) / (2/x)) = π/12

atan((1 − 2x + x²) / 2) = π/12

Take tangent of both sides.

(1 − 2x + x²) / 2 = tan(π/12)

1 − 2x + x² = 2 tan(π/12)

Half angle formula:

1 − 2x + x² = 2 sin(π/6) / (1 + cos(π/6))

1 − 2x + x² = 1 / (1 + √3 / 2)

1 − 2x + x² = 2 / (2 + √3)

Rationalize:

1 − 2x + x² = 2 (2 − √3) / (4 − 3)

1 − 2x + x² = 2 (2 − √3)

1 − 2x + x² = 4 − 2√3

Factor and solve:

(x − 1)² = 4 − 2√3

x − 1 = ±√(4 − 2√3)

x = 1 ± √(4 − 2√3)