Question

10. Rewrite with only sin x and cos x. (1 point)

Answer 1

Answer:

10. Option A is the correct answer.

11. Option D is the correct answer.

15. $cot\left ( x- \frac{\pi}{2} \right )=-tanx$ is verified.

Step-by-step explanation:

10. sin 2x - cos 2x

    sin 2x = 2 sinx cosx

    cos 2x  = cos²x-sin²x = 1-2sin²x

    sin 2x - cos 2x  = 2 sinx cosx - ( 1-2sin²x) = 2 sinx cosx - 1 + 2sin²x

    Option A is the correct answer.

11. sin 22.5°

     We have

             $sin\left ( \frac{\theta }{2} \right )=\sqrt{\frac{1-cos\theta }{2}}$

      $sin(22.5)=sin\left ( \frac{45 }{2} \right )=\sqrt{\frac{1-cos45 }{2}}=\sqrt{\frac{1-\frac{1}{\sqrt{2}} }{2}}=\sqrt{\frac{\sqrt{2}-1}{2\sqrt{2}}}\\\\sin(22.5)=\frac{1}{2}\sqrt{2-\sqrt{2}}$

Option D is the correct answer.

15. Verify $cot\left ( x- \frac{\pi}{2} \right )=-tanx$

 Substitute $x=\frac{\pi }{3}$

           $cot\left ( \frac{\pi}{3}-\frac{\pi }{2} \right )=cot\left ( \frac{-\pi}{6}\right )=-cot\left ( \frac{\pi}{6}\right )=-\sqrt{3}$

           $-tan\left ( \frac{\pi}{3} \right )=-\sqrt{3}$

   Hence equality verified.

Answer 2

10. Rewrite with only sin x and cos x.
sin 2x - cos 2x

sin2x = 2sinxcosx
cos2x = (cosx)^2 - (sinx)^2 = 2(cosx)^2 -1 = 1- 2(sinx)^2

sin2x- cos2x=2sinxcosx-(1- 2(sinx)^2=2sinxcosx-1+2(sinx)^2

sin2x- cos2x=2sinxcosx-1+2(sinx)^2

the answer is the letter b) 2 sin x cos2x - 1 + 2 sin2x

11. Find the exact value by using a half-angle identity. sin 22.5°Using the half angle formula you get:sin2(θ)=12[1−cos(2θ)]if θ=22.5° then 2θ=45°so you get:sin2(22.5°)=12[1−cos(45°)]sin2(22.5°)=12[1−√2/2]=2−√24and square root both sides:sin(22.5°)=±√2−√24=±0.382so 
sin(22.5°)=0.382the answer is the letter D) one half times the square root of quantity two minus square root of two
15. Verify the identity. 
cot x minus pi divided by two. = -tan x
Cot(x-pi/2)=-tan(x)
 sin(A − B) = sin A cos B − cos A sin B
sin(x – pi/2) = sin x cos (pi/2) − cos x sin (pi/2)=-cosx
 
cos(A − B) = cos A cos B − sin A sin B
cos(x− pi/2) = cos x cos pi/2 − sin x sin pi/2=-sinx
 Cot(x-pi/2)=cos(x-pi/2)/sin(x-pi/2)
= (-sinx)/(-cosx)=-tanx--------------ok