Prove the following identity: cos4x - sin4xcot2x = -1
Expert Answers
| Certified Educator
calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
We have to prove that cos 4x - sin 4x * cot 2x = -1
cos 4x - sin 4x * cot 2x = -1
use cos 2x = (cos x)^2 - (sin x)^2 and sin 2x = 2 sin x cos x and cot x = (cos x)/(sin x)
=> (cos 2x)^2 - (sin x)^2 - 2*(sin 2x)*(cos 2x)*(cos 2x)/(sin 2x)
=>(cos 2x)^2 - (sin x)^2 - 2*(cos 2x)*(cos 2x)
=> ( cos 2x)^2 - 2 ( cos 2x)^2 - ( sin 2x)^2
=> - (cos 2x)^2 - (sin 2x)^2
=> -1*[(cos 2x)^2 + (sin 2x)^2]
As (cos x)^2 + (sin x)^2 = 1
=> -1
Therefore we proved that
cos 4x - sin 4x * cot 2x = -1
Related Questions
We know that cos 4x = cos 2*(2x) = 2(cos 2x)^2 - 1
We'll write sin 4x = sin 2(2x) = 2 sin (2x)*cos (2x)
We'll write cot 2x = cos 2x/sin 2x
We'll re-write the identity:
2(cos 2x)^2 - 1 - 2 sin (2x)*cos (2x)*cos 2x/sin 2x = -1
We'll simplify and we'll get:
2(cos 2x)^2 - 1 - 2*cos 2x*cos 2x = -1
2(cos 2x)^2 - 1 - 2(cos 2x)^2 = -1
We'll add 2(cos 2x)^2 both sides:
2(cos 2x)^2 - 1 = 2(cos 2x)^2 - 1 q.e.d.
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.
Ask a Question
Popular Questions
- How do we derive the expansion for sin 3x?
-
1 educator answer
- Solve the equation 5sinx=4cosx
-
3 educator answers
- trigonometryGiven that sina + sinb =1 and cosa + cosb = 1/2 calculate cos(a-b) .
-
1 educator answer
- Prove that sin^4 x+cos^4 x+(sin^2 2x)/2=1
-
1 educator answer
- how to evaluate cos13pi/6 and sin37pi/4?
-
1 educator answer
- Calculate the expression sin^-1(sin(5pi/3))+tan^-1(tan(2pi/3))?
-
1 educator answer
- finding an anglewhat is the x angle if 2sin2x+square root 2=0?
-
1 educator answer
- Solve the equation tan^2x=8-8secx.
-
1 educator answer
Student Answers