Answer:
x= - cot²A
Step-by-step explanation:
part 1: tan (90° + A)* cot (180° - A) =
sin(90° + A)/cos(90° + A) * cos(180° - A)/sin(180° - A) =
(sin90°cosA + cos90°sinA)/(cos90°cosA+sin90°sinA)*(cos180°cosA+sin180°sinA)/(sin180°cosA + cos180°sinA) =
cosA/sinA * -cosA/-sinA = cos²A/sin²A
part 2: x*sec (90° +A)* cosec A = x* 1/cos(90° +A)* 1/sinA =
x* (1/cos90°cosA+sin90°sinA)*(1/sinA) = x*1/sinA*1/sinA = x/sin²A
part 3: x.cot A.tan (90° + A) = x*cosA/sinA*sin(90° + A)/cos(90° + A) =
x*cosA/sinA* (sin90°cosA+cos90°sinA)/(cos90°cosA+sin90°sinA) =
x*cosA/sinA* cosA/sinA = x* cos²A/sin²A
conclusion: part 1 + part 2 = part 3
cos²A/sin²A + x/sin²A = x* cos²A/sin²A
x/sin²A - x* cos²A/sin²A = - cos²A/sin²A
x - x*cos²A = - cos²A
x (1 - cos²A) = - cos²A
x * sin²A = - cos²A
x = -cot²A
