Prove the following identity: (tan^4) t + (tan^2) t + 1 = (1 - (sin^2) t * (cos^2) t) / (cos^4) t
- print Print
- list Cite
Expert Answers
Tushar Chandra
| Certified Educator
calendarEducator since 2010
write12,551 answers
starTop subjects are Math, Science, and Business
We have to prove that (tan t)^4 + (tan t)^2 + 1 = (1 - (sin t)^2* (cos t)^2) / (cos t)^4
The left hand side:
(tan t)^4 + (tan t)^2 + 1
=> (sin t)^4 / (cos t)^4 + (sin t)^2 / (cos t)^2 + 1
=> (sin t)^4 / (cos t)^4 + (cos t)^2*(sin t)^2 / (cos t)^4 + (cos t^4) / (cos t)^4
=> [(sin t)^4 + (cos t)^2*(sin t)^2 + (cos t^4)] / (cos t)^4
=> [ (sin t)^4 + (cos t )^4 + (cos t)^2*(sin t)^2]/(cos t)^4
=> [ ((sin t)^2 + (cos t)^2)^2 - 2*(cos t)^2*(sin t)^2 +(cos t)^2*(sin t)^2]/(cos t)^4
=> [ 1 - (cos t)^2*(sin t)^2]/(cos t)^4
which is the right hand side.
Therefore we have proved that (tan t)^4 + (tan t)^2 + 1 = (1 - (sin t)^2* (cos t)^2) / (cos t)^4
Related Questions
- Prove that : sin 3a+sin a+sin 5a/cos 3a+cos a+cos 5a=tan 3a
- 2 Educator Answers
- Prove the following identity: (sqrt (1 - (cos^2 ) t ) * sqrt ((sec^2) t - 1)) / cos t = (tan^2) t
- 1 Educator Answer
- I wish to prove cos^2x-sin^2x=2cos^2x-1.
- 3 Educator Answers
- What is sin(x+y) if sinx+cosy=1/4 and cos x+siny=1/2 ?
- 1 Educator Answer
- prove tan(a+b)=(tan a+tanb)/(1-tana*tanb)
- 1 Educator Answer