secxcscx(tan x + cot x) = 2 + tan^2 x + cot^2 x

Question

solve secxcscx(tanx + cot x) on the left hand side which equals 2 + tan^2 x + cot^2 x

Answer 1

First, note that the left-hand side simplifies as follows:

sec x csc x (tan x + cot x)

= (1/cos x) (1/sin x) [(sin x/cos x) + (cos x/sin x)]

= (1/cos x) (1/sin x) (sin x/cos x) + (1/cos x) (1/sin x)(cos x/sin x)

= 1/cos²x + 1/sin²x = sec²x + csc²x

Now use the fact that 1 + tan²x = sec²x and 1 + cot^2x = csc²x.

Please let me know if you need further clarification (i.e., more details on the last part).

Answer 2