SOLUTION
STEP 1: Pick one equation to work on first.
STEP 2: Write equation out on a different paper.
STEP 3: The equation in the numerator is CSC^4x - COT^4x. George notices that by expanding the equation, he'll be able to reduce with the equation in the denominator because they make 1. He'll expand it to (CSC²x-COT²x)(CSC²x+COT²x).
STEP 4: Reduce (CSC²x+COT²x) from the numerator and denominator.
STEP 5: By reducing (CSC²x+COT²x) from the numerator and denominator what George will be left with is CSC²x-COT²x +COT²x.
STEP 6: George notices that -COT²x +COT²x make a zero pair. Therefore, CSC²x is left.
STEP7: Now that George finished solving for one given side he could start on the other.
STEP 8: Write equation out on a different paper.
STEP 9: George notices that the equation in the denominator is also a Pythagorean identity where 1 + TAN² x = SEC²x, but in George's case it is rearranged into where TAN² x = SEC²x -1.
STEP 10: George also notices that SEC²x is a reciprocal identity where SECx = 1/ COS²x. He also notices that TAN²x is equal to SIN²x/COS²x. Now what George has is 1/ COS²x/ SIN²x/COS²x.
STEP 11: When a fraction is divided by another fraction what you do is you keep one of the fractions the same and multiply that by the reciprocal of the other fraction. So now what George has is ( 1/ COS²x)(COS²x/SIN²x).
STEP 12: When you multiply fractions you could reduce them diagonally because when you multiply it it's going to reduce anyway because it becomes 1 and when you multiply anything by one it's whatever you multiplied it to be.
STEP 13: Now what George is left with is 1/ SIN²x which is equal to CSC²x, it's a reciprocal identity.
STEP 14: Now that George has found the missing given sides he could now start working on finding what Z is equal to.
STEP 15: Since he knows the missing sides and the triangle is a right triangle, he could use the pythagorean theorem to solve for Z. The pythagorean theorem is a² + b² = c². So, now George could substitute the values CSC²x into a and b.
STEP 16: What George has now is (CSC²x)² + (CSC²x)² = Z².
STEP 17: CSC^4x + CSC^4x = Z². Once George adds them he gets 2CSC^4 = Z².
STEP 18: To get rid of the square on Z we'd have to square both sides.
STEP 19: So Z is equal to √2CSC^4
STEP 2: Write equation out on a different paper.
STEP 3: The equation in the numerator is CSC^4x - COT^4x. George notices that by expanding the equation, he'll be able to reduce with the equation in the denominator because they make 1. He'll expand it to (CSC²x-COT²x)(CSC²x+COT²x).
STEP 4: Reduce (CSC²x+COT²x) from the numerator and denominator.
STEP 5: By reducing (CSC²x+COT²x) from the numerator and denominator what George will be left with is CSC²x-COT²x +COT²x.
STEP 6: George notices that -COT²x +COT²x make a zero pair. Therefore, CSC²x is left.
STEP7: Now that George finished solving for one given side he could start on the other.
STEP 8: Write equation out on a different paper.
STEP 9: George notices that the equation in the denominator is also a Pythagorean identity where 1 + TAN² x = SEC²x, but in George's case it is rearranged into where TAN² x = SEC²x -1.
STEP 10: George also notices that SEC²x is a reciprocal identity where SECx = 1/ COS²x. He also notices that TAN²x is equal to SIN²x/COS²x. Now what George has is 1/ COS²x/ SIN²x/COS²x.
STEP 11: When a fraction is divided by another fraction what you do is you keep one of the fractions the same and multiply that by the reciprocal of the other fraction. So now what George has is ( 1/ COS²x)(COS²x/SIN²x).
STEP 12: When you multiply fractions you could reduce them diagonally because when you multiply it it's going to reduce anyway because it becomes 1 and when you multiply anything by one it's whatever you multiplied it to be.
STEP 13: Now what George is left with is 1/ SIN²x which is equal to CSC²x, it's a reciprocal identity.
STEP 14: Now that George has found the missing given sides he could now start working on finding what Z is equal to.
STEP 15: Since he knows the missing sides and the triangle is a right triangle, he could use the pythagorean theorem to solve for Z. The pythagorean theorem is a² + b² = c². So, now George could substitute the values CSC²x into a and b.
STEP 16: What George has now is (CSC²x)² + (CSC²x)² = Z².
STEP 17: CSC^4x + CSC^4x = Z². Once George adds them he gets 2CSC^4 = Z².
STEP 18: To get rid of the square on Z we'd have to square both sides.
STEP 19: So Z is equal to √2CSC^4
