Use identities to simplify each expression.

Question

1. (tan³(x)-sec²(x)tan(x))/cot(-x)
2. 1-(sec²(x)/tan²(x))
3. sin(x)+(cos²(x)/sin(x)

Answer 1

#1:  (tan³(x)-sec²(x)tan(x))/(cot(-x))

= (tan³(x)-sec²(x)tan(x))(-tan(x))   By cot (-x) = - tan (x) and by 1/cot x = tan x

= sec²(x)tan² (x) – tan⁴(x)             By distribution of -tan (x)

= tan² (x) (sec²(x)- tan² (x))          Pull out common tan² (x)

= tan² (x) (1 + tan² (x) - tan² (x))   By substitution of sec²(x) = 1 + tan² (x)        

= tan² (x)

#2:  1-(sec²(x)/tan²(x))

    = 1 – ((1 - tan²(x))/ tan²(x))     By sec² (x) = 1- tan² (x)

    = 1 – (1/ tan²(x)) – 1

    = - 1/tan²(x)

    = - cot (x)

#3: sin(x)+(cos²(x)/sin(x))

    = sin x ( 1 +(cos²(x)/sin²(x)) )

    = sin(x) (1 + cot²(x))

    = sin (x) csc²(x)                          By (1 + cot²(x)) = csc²(x)

    = sin (x)/ sin²(x)                          By csc²(x) = 1/sin²(x)

    = 1/ sin(x)

    = csc (x)