If tan25=a , than tan 205 _ tan115/ tan 245 + tan 335 =? , the given value is in degree?

tan25° = a
Then, cot25° = 1/a

Case 1 : If it is (tan205° - tan115°) / (tan245° + tan335°)
(tan205° - tan115°) / (tan245° + tan335°)
= [tan(180°+25°) - tan(90°+25°)] / [tan(270-25°) + tan(360°-25°)]
= [tan25° - cot(-25°)] / [cot25° + tan(-25°)]
= (tan25° + cot25°) / (cot25° - tan25°)
= [a + (1/a)] / [(1/a) - a]
= [(a²/a) + (1/a)] / [(1/a) - (a²/a)]
= (a² + 1) / (1 - a²)

Case 2 : If it is (tan205° + tan115°) / (tan245° + tan335°)
(tan205° + tan115°) / (tan245° + tan335°)
= [tan(180°+25°) + tan(90°+25°)] / [tan(270-25°) + tan(360°-25°)]
= [tan25° + cot(-25°)] / [cot25° + tan(-25°)]
= (tan25° - cot25°) / (cot25° - tan25°)
= [a - (1/a)] / [(1/a) - a]
= [(a²/a) - (1/a)] / [(1/a) - (a²/a)]
= (a² - 1) / (1 - a²)
= -(1 - a²) / (1 - a²)
= -1

Do you mean (tan(205)-tan(115)) / (tan(245)+tan(335)) ?
sum/difference formulas:
sin(α±β) = sinα·cosβ ± cosα·sinβ
cos(α±β) = cosα·cosβ ∓ sinα·sinβ

(tan205-tan115)/(tan245+tan335) = (a²+1)/(1-a²)

-1

tan(205) - tan(115) = = 2.6108...
tan(245) + tan(235) = 1.6781...

2.6108/1.6781 = 1.5558

Your equation does not have the variable "a= tan(25)"

tan(a + b) = sin(a + b) / cos(a + b) → recall: sin(a + b) = sin(a).cos(b) + cos(a).sin(b)

tan(a + b) = [sin(a).cos(b) + cos(a).sin(b)] / cos(a + b) → recall: cos(a + b) = cos(a).cos(b) - sin(a).sin(b)

tan(a + b) = [sin(a).cos(b) + cos(a).sin(b)] / [cos(a).cos(b) - sin(a).sin(b)] ← memorise this result



= tan(205)

= tan(25 + 180)

= tan(25)

= a ← memorise this result as (1)


= tan(115)

= tan(90 + 25) → recall the memorized result

= [sin(90).cos(25) + cos(90).sin(25)] / [cos(90).cos(25) - sin(90).sin(25)]

= [cos(25)] / [- sin(25)]

= - cos(25) / sin(25)

= - 1/tan(25)

= - 1/a ← memorise this result as (2)


= tan(245)

= tan(65 + 180)

= tan(65)

= tan(90 - 25) → recall the memorized result

= [sin(90).cos(- 25) + cos(90).sin(- 25)] / [cos(90).cos(- 25) - sin(90).sin(- 25)]

= [cos(- 25)] / [- sin(- 25)]

= [cos(25)] / [sin(25)]

= 1/tan(25)

= 1/a ← memorise this result as (3)



= tan(335)

= tan(155 + 180)

= tan(155)

= tan(180 - 25) → recall the memorized result

= [sin(180).cos(- 25) + cos(180).sin(- 25)] / [cos(180).cos(- 25) - sin(180).sin(- 25)]

= [- sin(- 25)] / [- cos(- 25)]

= [sin(25)] / [- cos(25)]

= - [sin(25)] / [cos(25)]

= - tan(25)

= - a ← memorise this result as (4)


= [tan(205) - tan(115)] / [tan(245) + tan(335)] → recall (1): tan(205) = a

= [a - tan(115)] / [tan(245) + tan(335)] → recall (2): tan(115) = - 1/a

= [a - (1/a)] / [tan(245) + tan(335)] → recall (3): tan(245) = 1/a

= [a - (1/a)] / [(1/a) + tan(335)] → recall (4): tan(335) = - a

= [a - (1/a)] / [(1/a) - a]

= - [(1/a) - a] / [(1/a) - a]

= - 1

That's 0

205-25=180
tan(205-25) = tan(180) = 0
tan(205-25) = (tan(205) - tan(25)) /(1+tan(205)tan(25)) = 0
tan(205)=tan(25) = a

115= 90+25
sin(115) = sin(90+25) = sin(90)cos(25)+cos(90)sin(25)
= (1)cos(25) + (0)sin(25)
= cos(25)
cos(115) = cos(90+25) = cos(90)cos(25)-sin(90)sin(25)
= (0)cos(25) - 1 (sin(25))
= -sin(25)
tan(115) = sin(115)/cos(115) = cos(25)/(-sin(25)) = -cot(25) = -1/tan(25) = -1/a

tan(205)=a
tan(115) =-1/a

245 = 270-25
sin(245) = sin(270-25) = sin(270)cos(25)-cos(270)sin(25)
= (-1)cos(25) - (0) sin(25)
= -cos(25)

cos(245)= cos(270-25) = cos(270)cos(25)+sin(270)sin(25)
= (0)cos(25) - (1) sin(25)
= -sin(25)

tan(245) = sin(245)/cos(245) = -cos(25)/sin(25) = cot(25) = 1/a
tan(245)=1/a

335= 360-35
tan(335) = tan(360-25) = (tan(360)-tan(25))/(1+tan(360)tan(25))
= (0-tan(25))/(1+0) = -tan(25) =-a

tan(205)=a
tan(115)=-1/a
tan(245) = 1/a
tan(335) = -a

(tan(205) -tan(115)) /(tan(245)+tan(335)) = (a -(-1/a)) / (1/a -a)
= (a +1/a) /(1/a-a)
= (a^2+1) /(1-a^2)

-1

[ a - 1 / a ] / [ 1 / a - a] = - 1