Expert: Sherman D. - 1/18/2008

Question
QUESTION: 1   /    csc^2x (1-cosx)  = 1+cosx



ANSWER: if you are asking

1/(csc(x)^2 * (1 - cos(x))) = 1 + cos(x)

1/((1/sin(x)^2)(1 - cos(x)))
1/(((1 - cos(x))/(sin(x)^2)))
1/((1 - cos(x))/(1 - cos(x)^2)))
1/((1 - cos(x))/((1 - cos(x))(1 + cos(x))))
1/(1/(1 + cos(x)))

(1/1)/(1/(1 + cos(x))
(1/1)*(1 + cos(x))
1 + cos(x)

1/((csc(x)^2)(1 - cos(x))) = 1 + cos(x)

---------- FOLLOW-UP ----------

QUESTION: 1/cotx (1/tanx + 1/cotx) =  sec^2x

ANSWER: IF YOU ARE SAYING

(1/cot(x))((1/tan(x)) + (1/cot(x))) = sec(x)^2

(1/cot(x)) = tan(x)

tan(x)((1/tan(x)) + tan(x))
1 + tan(x)^2

1 + (sin(x)/cos(x))^2
(cos(x)^2 + sin(x)^2)/cos(x)^2
1/cos(x)^2
sec(x)^2

so

(1/cot(x))((1/tan(x)) + (1/cot(x))) = sec(x)^2

---------- FOLLOW-UP ----------

QUESTION: sin^2x (1/sin^2x + 1/cos^2x) = sec^2x

Answer
sin(x)^2((1/sin(x)^2) + (1/cos(x)^2))) = sec(x)^2
1 + (sin(x)/cos(x))^2
1 + tan(x)^2

so

sin(x)^2((1/sin(x)^2) + (1/cos(x))^2))) = sec(x)^2

info found at http://math2.org/math/trig/identities.htm

if you wonder why i put "IF THIS IS WHAT YOU MEAN", the reason why is because you should put separate values in parenthesis.

another way to work your problem out

sin(x)^2((1/sin(x)^2) + (1/cos(x)^2)))
sin(x)^2((cos(x)^2 + sin(x)^2)/(sin(x)^2cos(x)^2))
sin(x)^2(1/(sin(x)^2cos(x)^2)))
1/cos(x)^2
sec(x)^2