simplify the following 3 solutions 1a sin x csc x sin x 1b

simplify the following 3 solutions

1a. sin x * (csc x - sin x)

1b. cot^2 x - csc^2 x

1c. (5-5sin x) (3+3sin x)

Solution

1a sinx (cscx -sinx)

we know cscx = 1/sinx

So,sinx (cscx -sinx) = sinx*1/sinx - sinx*sinx

= 1 - sin^2x

= cos^2x

b) cot^2x - csc^2x

we know cotx = cosx/sinx and cscx = 1/sinx

So, cot^2x - csc^2x = cos^2x/sin^2x - 1/sin^2x

= (cos^2x -1)/sin^2x

= -sin^2x/sin^2x

= -1

1c.    (5-5sin x) (3+3sin x)

expand the bracket terms:

15 +15sinx -15sinx -15sin^2x

15 -15sin^2x

15( 1 -sin^2x)

15cos^2x