Help me is the answer sec?

Question

Which of the following is equivalent to sin(theta) * (tan(theta) + cot(theta))' ?

A) (cos(theta) + 1)/(cos(theta) - 1);

B)cos(theta)

C)sin(theta)

D) sec(theta)

E)tan(theta)

F)None of the above

Answer 1

You are correct; the answer is sec(theta). To see why, first remember that cot = 1 / tan, and therefore tan + cot = tan + (1/tan) = (tan^2 + 1) / tan. Then, since tan^2 + 1 = sec^2, we can reduce sin * (tan + cot) to sin * sec^2 / tan. Finally, since sin / cos = tan, sin / tan = cos, and sin * (tan + cot) = cos * sec^2, which is clearly equal to sec because cos = 1 / sec.

Answer 2

sinx(tanx + cotx) = sinx(sinx/cosx + cosx/sinx)

= sin^2(x)/cosx + cosx

= sin^2(x)/cosx + cos^2(x)/cosx

= (sin^2(x) +cos^2(x))/cosx

= 1/cosx

= secx

option D) is correct