Question

Simplify: tan⁡(x)⋅(csc⁡(x)−sin⁡(x)
a) tan⁡(x)
b) sin⁡(x)
c) 1
d) cos⁡(x)

Answer 1

To simplify tan(x)(csc(x) - sin(x)), rewrite csc(x) as 1/sin(x), then tan(x)/sin(x) becomes cos(x), resulting in the simplified expression of cos(x) - sin^2(x).

Step-by-step explanation:

To simplify the expression tan(x)(csc(x) - sin(x)), we first need to understand the trigonometric identities involved. The cosecant function, denoted as csc(x), is the reciprocal of the sine function. Therefore, csc(x) can be rewritten as 1/sin(x). Using this knowledge, we can rewrite the expression as:

tan(x)(1/sin(x) - sin(x))

Simplifying further, tan(x) divided by sin(x) gives us cos(x) because tan(x) = sin(x)/cos(x). Hence:

cos(x) - sin^2(x)

Since sin^2(x) is just another way of writing (sin(x))^2, we can say that the expression simplifies to cos(x) - sin^2(x).