Simplify and write the trigonometric expression in terms of sine and cosine:

Question

Simplify and write the trigonometric expression in terms of sine and cosine:

Answer 1

Cotangent is an odd function therefore cot(-x) = -cot(x) also cot(x) = cos(x)/sin(x) therefore:

cot(-x) = -cos(x)/sin(x)

Cosine is an even function therefore:

cos(-x) = cos(x)

Sine is an odd function therefore:

sin(-x) = -sin(x)

So, putting these together:

cot(-x)cos(-x) + sin(-x) = [-cos(x)/sin(x)][cos(x)] - sin(x)

Simplifying, we get

cot(-x)cos(-x) + sin(-x) = -cos²(x)/sin(x) - sin(x)

Then, getting a common denominator:

cot(-x)cos(-x) + sin(-x) = -cos²(x)/sin(x) - sin²(x)/sin(x)

cot(-x)cos(-x) + sin(-x) = -1[cos²(x)/sin(x) + sin²(x)/sin(x)]

cot(-x)cos(-x) + sin(-x) = -1[cos²(x) + sin²(x)]/sin(x)

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

Since this equals -1/f(x) then f(x) = sin(x)