help me..tq vm..
-
tan^2 x + cot^2 x
= sin^2 x / cos^2 x + cos^2 x / sin^2 x
= (sin^4 x + cos^4 x) / sin^2 x cos^2 x
= ((1 - cos^2 x)^2 + cos^4 x) / sin^2 x cos^2 x
= (1 - 2 cos^2 x + cos^4 x + cos^4 x) / sin^2 x cos^2 x
= (1 - 2 cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x ; (use 1 = sin^2 x + cos^2 x)
= (sin^2 x + cos^2 x - 2 cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x
= (sin^2 x - cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x
= 1/cos^2 x - 1/sin^2 x + 2 cos^2 x / sin^2 x ; ( cos^2 x = 1 - sin^2 x)
= 1/cos^2 x - 1/sin^2 x + 2 (1 - sin^2 x)/ sin^2 x
= 1/cos^2 x - 1/sin^2 x + 2 /sin^2 x - 2 sin^2 x/ sin^2 x
= 1/cos^2 x + 1/sin^2 x - 2
= sec^2 x + csc^2 x - 2
= sin^2 x / cos^2 x + cos^2 x / sin^2 x
= (sin^4 x + cos^4 x) / sin^2 x cos^2 x
= ((1 - cos^2 x)^2 + cos^4 x) / sin^2 x cos^2 x
= (1 - 2 cos^2 x + cos^4 x + cos^4 x) / sin^2 x cos^2 x
= (1 - 2 cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x ; (use 1 = sin^2 x + cos^2 x)
= (sin^2 x + cos^2 x - 2 cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x
= (sin^2 x - cos^2 x + 2 cos^4 x) / sin^2 x cos^2 x
= 1/cos^2 x - 1/sin^2 x + 2 cos^2 x / sin^2 x ; ( cos^2 x = 1 - sin^2 x)
= 1/cos^2 x - 1/sin^2 x + 2 (1 - sin^2 x)/ sin^2 x
= 1/cos^2 x - 1/sin^2 x + 2 /sin^2 x - 2 sin^2 x/ sin^2 x
= 1/cos^2 x + 1/sin^2 x - 2
= sec^2 x + csc^2 x - 2