Question

Simplify the expression.

cos x cotx + sin x

a. 0
b. csc x
c. tanx
d. sec x

Answer 1

b. csc x

Explanation:

To simplify the expression;

cos x cotx + sinx

But cot x =
`(cos x)/(sinx)`

Substitute cot x =
`(cos x)/(sinx)` into the initial expression

cos x cotx + sinx

=
`cos x. (cos x)/(sinx) + sin x`

=
`(cos^(2) x)/(sinx) + sin x`

=
`(cos^(2) x + sin^(2)x )/(sin x)` But
`sin^(2)x + cos^(2)x = 1`

=
`(1)/(sinx)` But
`(1)/(sinx)`= cosec x

= cosec x

Therefore, cos x cotx + sinx = cosec x

Answer 2

b. csc x

Explanation:

cos x cotx + sin x

We know that cot = cos/sin

cos x cos x/ sin x + sin x

cos^x /sin x + sin x

Getting a common denominator of sin x

cos^2 x /sin x + sin x * sin x / sin x

cos ^2 x / sin x + sin^2 x / sin x

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

We know that cos ^2 x + sin^2 x = 1

1 / sin x

We know that 1 / sin x = csc x

csc x