Prove that: (cos^3x - sin^3x)/(cosx - sinx) = 1 + cosx*sinx is always true.
- print Print
- list Cite
Expert Answers
justaguide
| Certified Educator
calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
We have to prove that [(cos x)^3 - (sin x)^3]/[cos x - sin x]= 1 + cos x * sin x
Starting with the left hand side:
(cos x)^3 - (sin x)^3 / cos x - sin x
use a^3 - b^3 = (a - b)(a^2 + ab + b^2)
=> [(cos x - sin x)[(cos x)^2 + cos x * sin x + (sin x)^2]]/( cos x - sin x)
cancel (cos x - sin x)
=> [(cos x)^2 + cos x * sin x + (sin x)^2]
Use (cos x)^2 + (sin x)^2 = 1
=> 1 + cos x * sin x
This is the right hand side.
This proves that [(cos x)^3 - (sin x)^3]/[cos x - sin x] = 1 + cos x * sin x
check Approved by eNotes Editorial
Related Questions
- How to prove the identity sin^4x - cos^4x / sin^3x - cos^3x = Sinx + cosx / 1 + sinxcosx
- 1 Educator Answer
- Prove the identity `{1-sinx}/cosx=cosx/{1+sinx}`
- 1 Educator Answer
- Prove cosx/(1-tanx) - cosx = sin x - sinx/(1-cotx)
- 1 Educator Answer
- Prove the identity: (cosx/1-sinx) - (cosx/1+sinx) = 2tanx ------------------------------- Thank...
- 1 Educator Answer
- How to prove the identity `sin^2x + cos^2x = 1` ?
- 3 Educator Answers
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.