Question

if tan 7.5° *tan 52.5° * tans 67.5° =
$\sqrt{a} - b$
then a +b is ​

Answer 1

Gɪᴠᴇɴ :-

• tan 7.5° *tan 52.5° * tan 67.5° = (√a) - b

Tᴏ Fɪɴᴅ :-

• (a + b)

Fᴏʀᴍᴜʟᴀ ᴜsᴇᴅ :-

• tanA * tan(60 - A) * tan(60 + A) = 3 * tan(3A)
• tan(A/2) = √[(1 - cosA)/(1 + cosA)

Sᴏʟᴜᴛɪᴏɴ :-

→ tan 7.5° *tan 52.5° * tan 67.5°

→ tan 7.5° * tan(60 - 7.5°) * tan(60 + 7.5°)

comparing with tanA * tan(60 - A) * tan(60 + A) now , we get,

→ 3 * tan(3 * 7.5°)

→ 3 * tan(22.5°)

____________________

Now, we have to Find value of tan(22.5°) = tan(45°/2) .

using tan(A/2) = √[(1 - cosA)/(1 + cosA)] we get,

→ tan(45°/2) = √[(1 - cos45°)/(1 + cos45°)]

Putting value of cos45° = (1/√2) now, we get,

→ √[(1 - (1/√2))/(1 + (1/√2))]

→ √[ (√2 - 1) / (√2 + 1) ]

Rationalizing Now,

→ √[ {(√2 - 1) / (√2 + 1)} * {(√2 - 1) / (√2 - 1)} ]

→ √[ (√2 - 1)² / (2 - 1) ]

→ (√2 - 1)

_____________________

Therefore,

→ 3(√2 - 1) = (√a) - b

→ 3√2 - 3 = (√a) - b

→ √(9*2) - 3 = (√a) - b

→ √(18) - 3 = (√a) - b

comparing Now, we get,

☛ a = 18

☛ b = 3

Hence,

☞ a + b = 18 + 3 = 21 (Ans.)

____________________________