Choose the one alternative that best completes the statement

Choose the one alternative that best completes the statement or answer the question. Complete the identity. Sin 2x sin 5x cos 2x cos 5x = ? cos^2 3x - cos^2 7x/4 cis^2 20x sin^2 20x/4 cos^27x + cos^2 3x/4 cos 2x - cos 8x/cos 2x + cos 8x = ? tan 3x tan 5x cot 5x 0 -tan 5x Find all solutions of the equation. 2 cos x - 1 = 0 x = pi/3 + 2n pi or x = 5pi/3 + 2n pi x = 4pi/3 + n pi or x = 5 pi/6 + n pi x = pi/3 + npi or x = 5 pi/3 + n pi x= 4 pi/3 + 2npi or x = 5 pi/6 + 2n pi tan x sec x = - 2 tan x x = pi/3 + n pi ot x = 5 pi/3 + n pi or x = n pi x = pi/3 + 2n pi or x = 5 pi/3 + 2n pi or x = n pi x = 2 pi/3 + n pi or x = 4 pi/3 + n pi or x = n pi x = 2 pi/3 + 2n pi or x = 4 pi/3 + 2n pi or x = n pi Solve the equation on the interval [0, 2 pi). Cos 2x = squareroot 3/2 pi/2 3 pi/2 pi/12, 11 pi/12, 13 pi/12, 23 pi/12 pi/6, 11 pi/6 Solve the equation on the interval [0, 2pi). -tan^2x sin x = -tan^2x pi, 2 pi 0, 3 pi/2 0, pi pi, 3 pi/2 Solve the equation on the interval [0, 2 pi). Cos 2x = squareroot 2- cos 2x pi/4, 3pi/4, 5pi/4, 7pi/4 0, 2 pi/3, pi, 4 pi/3 pi/8, 7 pi/8, 9 pi/8, 15 pi/8 no solution

Solution

1) sin 2x sin 5x cos 2x cos 5x

let 2x = a

5x = b

sin a sin b = -1/2 [ cos (a+b) -cos (a-b) ]

-1/2 [ cos 7x - cos -3x ]

cos a cos b = 1/2 [sin (a+b) - sin (a-b) ]

cos 2x cos 5x = 1/2 [ sin 7x - sin (-3x) ]

-1/2 [ cos 7x - cos -3x ] * 1/2 [ sin 7x - sin (-3x) ] = 1/4 [ cos^2 3x - cos^2 7x ] ( option a)

2) cos 2x - cos 8x / cos 2x + cos 8x

cos 2x = cos ( 5x - 3x )

cos 8x = cos ( 5x + 3x )

plugging the values back

cos ( 5x - 3x ) - cos ( 5x + 3x ) / cos ( 5x - 3x ) + cos ( 5x + 3x )

applying cos (a-b) and cos (a+b) formula

cos (a-b) = cos a cos b + sin a sin b

cos (a+b) = cos a cos b - sin a sin b

cos ( 5x - 3x ) - cos ( 5x + 3x ) / cos ( 5x - 3x ) + cos ( 5x + 3x )

cos 5x cos 3x + sin 3x sin 5x - [ cos 5x cos 3x -sin 5x sin 3x ] / cos 5x cos 3x + sin 3x sin 5x + [ cos 5x cos 3x -sin 5x sin 3x ]

sin^2 5x sin^3x / cos^2 5x cos^2 3x

sinx / cos x= tan x

= tan 5x tan 3x

3) 2 cos x - 1 = 0

adding 1 one both sides

2 cos x = 1

dividing both sides by 2

cos x = 1/2

x = pi/3 + 2pi*n

or

x = 5pi/3 + 2 pi* n

(option a)

4) tan x sec x = - 2 tan x

adding 2 tan x on both sides

tan x sec x + 2 tan x = 0

taking tan x as gcf

tan x( sec x + 2 ) = 0

tan x = 0

sec x + 2 = 0

x = pi*n

or

x = 2pi/3 + 2pi*n , 4pi/3 + 2pi*n

option d