Verified answer

sin(x) = 7/17

cos(x) = sqrt(1 - sin(x)^2) = sqrt(1 - 49/289) = sqrt(240 / 289) = sqrt(16 * 15 / 289) = (4/17) * sqrt(15)

sin(x/2) =>

sqrt((1/2) * (1 - cos(x))) =>

sqrt((1/2) * (1 - (4/17) * sqrt(15))) =>

sqrt((2/4) * (17 - 4 * sqrt(15)) / 17) =>

sqrt((34 / (4 * 17^2)) * (17 - 4 * sqrt(15))) =>

sqrt(34 * (17 - 4 * sqrt(15))) / 34

cos(x/2) =>

sqrt((1/2) * (1 + cos(x))) =>

sqrt((2/4) * (1 + (4/17) * sqrt(15))) =>

(1/17) * (1/2) * sqrt(2 * 17 * (17 + 4 * sqrt(15))) =>

sqrt(34 * (17 + 4 * sqrt(15))) / 34

tan(x/2) =>

sin(x/2) / cos(x/2) =>

sqrt(34 * (17 - 4 * sqrt(15))) / sqrt(34 * (17 + 4 * sqrt(15))) =>

sqrt((17 - 4 * sqrt(15)) / (17 + 4 * sqrt(15))) =>

(17 - 4 * sqrt(15)) / sqrt(289 - 16 * 15)) =>

(17 - 4 * sqrt(15)) / sqrt(49) =>

(17 - 4 * sqrt(15)) / 7

sin(x) = 7/17 → where 0° < x < 90° → cos(x) > 0

cos²(x) + sin²(x) = 1

cos²(x) = 1 - sin²(x)

cos²(x) = 1 - (7/17)²

cos²(x) = 1 - (49/289)

cos²(x) = (289/289) - (49/289)

cos²(x) = (289 - 49)/289

cos²(x) = 240/289 → recall cos(x) > 0

cos(x) = (√240)/(√289)

cos(x) = (√240)/17

cos(x) = [√(16 * 15)]/17

cos(x) = (4√15)/17

tan(x) = sin(x)/cos(x)

tan(x) = (7/17) / [(4√15)/17]

tan(x) = 7/(4√15)

Do you know this identity:

cos(2x) = 2cos²(x) - 1

Adapt this result to your case:

cos(x) = 2cos²(x/2) - 1

2cos²(x/2) = cos(x) - 1 → recall: cos(x) = (4√15)/17

2cos²(x/2) = [(4√15)/17] - 1

2cos²(x/2) = [(4√15)/17] - (17/17)

2cos²(x/2) = (4√15 - 17)/17

cos²(x/2) = (4√15 - 17)/34

cos(x/2) = √[(4√15 - 17)/34]

Do you know this identity:

sin(2x) = 2sin(x).cos(x)

Adapt this result to your case:

sin(x) = 2sin(x/2).cos(x/2)

2sin(x/2) = sin(x)/cos(x/2) → recall: sin(x) = 7/17

2sin(x/2) = (7/17)/cos(x/2) → recall: cos(x/2) = √[(4√15 - 17)/34]

2sin(x/2) = (7/17)/√[(4√15 - 17)/34]

2sin(x/2) = (7/17) * 1/√[(4√15 - 17)/34]

sin(x/2) = (7/34) * 1/√[(4√15 - 17)/34]

Do you know this identity:

tan(2x) = 2tan(x) / [1 - tan²(x)]

Adapt this result to your case:

tan(x) = 2tan(x/2) / [1 - tan²(x/2)]

Faster with:

tan(x/2) = sin(x/2) / cos(x/2)

tan(x/2) = { (7/34) * 1/√[(4√15 - 17)/34] } / { √[(4√15 - 17)/34] }

tan(x/2) = { (7/34) * 1/√[(4√15 - 17)/34] } * { 1/√[(4√15 - 17)/34] }

tan(x/2) = (7/34) * 1/[(4√15 - 17)/34]

tan(x/2) = 7/[(4√15 - 17)/34²]

tan(x/2) = (7 * 34²)/(4√15 - 17)