Find tan 75 using both sum and difference formulas. Thank you!?

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)
tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

tan75°
= tan(45° + 30°)
= (tan45° + tan30°) / (1 - tan45° tan30°)
= [1 + (1/√3)] / [1 - 1 × (/√3)]
= [(√3 + 1)/√3] / [(√3 - 1)/√3]
= (√3 + 1) / (√3 - 1)
= [(√3 + 1) / (√3 - 1)] × [(√3 + 1) / (√3 + 1)]
= (3 + 1 + 2√3) / (3 - 1)
= 2 + √3

tan75°
= tan(120° - 45°)
= (tan120° - tan45°) / (1 + tan120° tan45°)
= [tan(180° - 60°) - tan45°] / [1 + tan(180° - 60°) tan45°]
= [(-tan60°) - tan45°] / [1 + (-tan60°) tan45°]
= [-√3 - 1] / [1 + (-√3) × 1]
= (-√3 - 1) / (1 - √3)
= (√3 + 1) / (√3 - 1)
= 2 + √3

tan 75 
= tan (45 + 30)
= (√ 3 + 1) / (√3 − 1) 
= 2.73 / 0.73 
= 3.73 
 . 

Do you know this identity?

cos(a + b) = cos(a).cos(b) - sin(a).sin(b) → suppose that: a = 45 and suppose that: b = 30

cos(45 + 30) = cos(45).cos(30) - sin(45).sin(30)

cos(75) = [(√2)/2 * (√3)/2] - [(√2)/2 * (1/2)]

cos(75) = [(√6)/4] - [(√2)/4]

cos(75) = (√6 - √2)/4


Do you know this identity?

sin(a + b) = sin(a).cos(b) + cos(a).sin(b) → suppose that: a = 45 and suppose that: b = 30

sin(45 + 30) = sin(45).cos(30) + cos(45).sin(30)

sin(75) = [(√2)/2 * (√3)/2] + [(√2)/2 * (1/2)]

sin(75) = [(√6)/4] + [(√2)/4]

sin(75) = (√6 + √2)/4


tan(75) = sin(75)/cos(75)

tan(75) = [(√6 + √2)/4] / [(√6 - √2)/4]

tan(75) = (√6 + √2)/(√6 - √2)

tan(75) = (√6 + √2)²/[(√6 - √2).(√6 + √2)]

tan(75) = (6 + 2√12 + 2)/[6 - 2]

tan(75) = (8 + 2√12)/4

tan(75) = (8 + 4√3)/4

tan(75) = 2 + √3

tan 75 degrees = 3.73205080757

tan 75 degrees = 2 + √ 3 
Explanation: 
tan ( 75 )  = tan ( 30 + 45 ) 
now  
tan ( A + B ) = (tan A + tan B) / (1 − tan A tan B )
tan ( 75 ) = tan ( 30 + 45 ) 
= (tan 30 + tan 45) / (1 − tan 30 tan 45)

tan 30 = √3/3 
,
tan 45 = 1 

∴ tan 75 = (√3/3 + 1) / (1 − √3/3)

= (√3 + 3) / (3 − √3) × (3 + √3) / (3 + √3)
= 3√3 + 3 + 9 + 3 √ 3 6 
= 12 + 6√3 /6 
cancel the 6 
= 2 + √ 3

I assume all angles are in degrees:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)*tan(B)) 
tan(30 + 45) = (tan(30) + tan(45)) / (1 - tan(30)*tan(45))
tan(75) = (sqrt(1/3) + 1) / (1 - sqrt(1/3)*1)
tan(75) = (sqrt(1/3) + 1) / (1 - sqrt(1/3))
tan(75) = (sqrt(3) + 1) / (sqrt(3) - 1)
tan(75) = sqrt(3) + 2