TIGONOMETRIC ( I HAVE EXAM TODAY!!!)?

Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base
Cosec A = Hypotenuse/Perpendicular
Sec A = Hypotenuse/Base
Cot A = Base/Perpendicular

Now Imagine the 4 quadrants formed by x and y axes

If the point is in first quadrant P=+ and B=+ therefore H=+
Sin, Cos, Tan are + therefore Cosec, Sec, Cot are + as well.

If the point is in second quadrant P=+ and B=- therefore H=+
Sin + , Cos -, Tan - therefore Cosec +, Sec -, Cot - .

If the point is in third quadrant P=- and B=- therefore H=+
Sin - , Cos -, Tan + therefore Cosec -, Sec -, Cot + .

If the point is in 4th quadrant P=- and B=+ therefore H=+
Sin - , Cos +, Tan - therefore Cosec -, Sec +, Cot - .

Now see
90-A is the angle in first quadrant
180-A is the angle in 2nd quadrant
270-A is the angle in 3rd quadrant
360-A is the angle in 4th quadrant
(provided A is less than 90 degrees otherwise decide the quadrant first)

Now if it is 90 or 270 degrees sin will become sine will become cos
cos will become sine
tan will become cot
cot will become tan
sec will become cosec
cosec will become sec

If it is 180 or 360 everything will remain same

Now
cot(90-A) =tan A
since it is 90 degrees cot will become tan
and since 90-A means first quadrant
Tan is positive.

another thumb rule
sin(-A)=-sinA
cos(-A)=cos A
tan(-A)=-tan A
-A means 4th quadrant and
If the point is in 4th quadrant P=- and B=+ therefore H=+
Sin - , Cos +, Tan - therefore Cosec -, Sec +, Cot - .

therefore tan(A-180) can be writen as tan-(180-A)
=-tan(180-A)
since it is 180 tan will remain tan
and it is 2nd quadrant
and
If the point is in second quadrant P=+ and B=- therefore H=+
Sin + , Cos -, Tan - therefore Cosec +, Sec -, Cot - .
therefore
it will become -(-tanA)
=tanA

cot(90-A) = tanA
tan(A-180) = tanA
sin(A-90) = cosA

[tan(270-A) + cot(A+180)] / [1 + tan(A-90)cot(180-A)]
= [tan(90-A) + tan(90-A)] / [1 + tan(90-A)tan(90-A)]
= 2tan(90-A) / (1 + tan²(90-A))
= 2cot(A) / (1 + cot²(A))
= 2cot(A) / csc²(A)
= [2(cos(A)/sin(A)] × sin²(A)
= 2sin(A)cos(A)
= sin(2A)

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TAN(270-A)=COT(A)
COT(A+180)=COT(A)
TAN(A-90)= -COT(A)
COT(180-A)=-TAN(A)
COT(A)*TAN(A)=1
so we have:
cot(A)+cot(A)/1+(-cot(A))(-tan(A))
2cot(A)/1+1
cot(A)

The original expression=
[cotA+cotA]/[1-cot^2(A)]=
2tanA/[tan^2(A)-1]

cot(90*-A)=tanA

tan(A-180*)=
-tan(180*-A)=
tanA

sin(A-90*)=
-sin(90*-A)=
-cosA