SOLUTION: prove (1 / tan^2θ ) + (1 / 1+ cot^2θ) = 1 and (sinθ + cosθ)(tanθ + cotθ) = cscθ + secθ . plz help thanks

Algebra ->  Trigonometry-basics -> SOLUTION: prove (1 / tan^2θ ) + (1 / 1+ cot^2θ) = 1 and (sinθ + cosθ)(tanθ + cotθ) = cscθ + secθ . plz help thanks      Log On


   

Question 217308: prove
(1 / tan^2θ ) + (1 / 1+ cot^2θ) = 1
and
(sinθ + cosθ)(tanθ + cotθ) = cscθ + secθ .
plz help thanks
Answer by chibisan(131) About Me  (Show Source):
You can put this solution on YOUR website!
1)
L.H.S
1/1+tan^2θ + 1/1+cot^2θ
1/sec^2θ + 1/cosec^2θ
cos^2θ + sin^2θ = 1 (proven)
2)
L.H.S
(sinθ+cosθ)(tanθ+cotθ)
= sinθ(sinθ/cosθ) + cosθ + sinθ + cosθ(cosθ/sinθ)
= sin^2θ/cosθ + cosθ + sinθ + cos^2θ/sinθ
make into single fraction
= (sin^2θ + cos^2θ/cosθ) + (cos^2θ + sin^2θ/sinθ)
note : sin^2θ + cos^2θ = 1
= 1/cosθ + 1/sinθ
= secθ + cosecθ (proven)