If sinϴ+cosϴ = √2 cosϴ prove that cosϴ-sinϴ = √2 sinϴ, (ϴ is an acute angle)
- print Print
- list Cite
Expert Answers
briefcaseTeacher (K-12)
calendarEducator since 2011
write3,158 answers
starTop subjects are Math, Science, and Business
Given `sin theta + cos theta = sqrt(2)cos theta` prove `cos theta - sin theta = sqrt(2)sin theta`
`sin theta + cos theta = sqrt(2)cos theta` square both sides
`sin^2theta + cos^2 theta + 2sintheta costheta=2cos^2theta`
`sin^2theta - cos^2theta + 2sinthetacostheta=0`
`-sin^2theta + cos^2theta -2sinthetacostheta=0` Add `2sin^2theta` to both sides
`sin^2theta+cos^2theta-2sinthetacostheta=2sin^2theta`
`(costheta-sintheta)^2=2sin^2theta`
`costheta-sintheta=sqrt(2)sintheta` as required.
Related Questions
- prove that (sinθ + cosecθ)^2 + (cosθ + secθ)^2=7+tan^2θ+cot^2θ
- 2 Educator Answers
- If 3 cosθ = 5 sinθ, then the value of (5 sinθ - 2sec^3θ + 2 cosθ)/(5sinθ + 2sec^3θ - 2cosθ)
- 2 Educator Answers
- If tan Θ/2 = t, express each of the following in terms of t (sinΘ+sinΘ/2)/1+cosΘ+cosΘ/2 = t...
- 1 Educator Answer
- If x `sin^3 θ` `theta` + y `cos^3 θ ``theta` = sin θ cos θ, and x sin θ = y cos θ, prove that...
- 1 Educator Answer
- Prove (Show complete solution and explain): (1-Sinθ)/Cosθ = Cosθ/(1+Sinθ)
- 1 Educator Answer
calendarEducator since 2011
write562 answers
starTop subjects are Math, Science, and Business
Divide by `costheta` to get
` tantheta+1=sqrt(2)`
so
`tantheta=sqrt(2)-1`
`1/tantheta=cottheta`
Take the reciprocal of both sides
`cottheta=1/(sqrt(2)-1)`
`1/(sqrt(2)-1)*(sqrt(2)+1)/(sqrt(2)+1)=(sqrt(2)+1)/(2-1)=sqrt(2)+1`
So
` cottheta=sqrt(2)+1`
`costheta/sintheta=sqrt(2)+1`
`costheta/sintheta-1=sqrt(2)`
Multiply both sides by `sintheta`
`costheta-sintheta=sqrt(2)sintheta`
Which is what we were supposed to prove.
cosϴ +sinϴ = √2 cosϴ (given)
squaring on the both sides , we get
cos2ϴ + sin2ϴ + 2cosϴsinϴ = 2 cos2ϴ
sin2ϴ + 2cosϴsinϴ = cos2ϴ
cos2ϴ - 2cosϴsinϴ = sin2ϴ
cos2ϴ - 2cosϴsinϴ + sin2ϴ = 2 sin2ϴ (adding sin2ϴ on both sides)
(cosϴ - sinϴ)2 = 2 sin2ϴ
cosϴ-sinϴ = √2 sinϴ
Hence proved!!
Student Answers