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Help solving trig proofs?
[(cos^2)x+1]/(sin^2)x = 2(csc^2)x - 1
also...
(cscx)/(tanx+cotx) = cosx
I had four of these to do and can't figure out these two.
1 Answer
- ?Lv 71 decade agoFavorite Answer
1)
Note that cos^2 (x) + sin^2 (x) = 1
Thus, cos^2 (x) = 1 - sin^2 (x)
(cos^2 (x) + 1) / sin^2 (x)
= [1 - sin^2 (x) + 1] / sin^2 (x)
= [2 - sin^2 (x)] / sin^2 (x)
= [2 / sin^2 (x)] - [sin^2 (x) / sin^2 (x)]
= 2csc^2 (x) - 1
Note that csc (x) = 1 / sin (x)
2)
csc (x) / (tan (x) + cot (x))
= (1 / sin (x)) / [(sin (x) / cos (x)) + (cos (x) / sin (x))]
join the denominator into one term
= (1 / sin (x)) / [(sin^2 (x) + cos^2 (x)) / (sin (x) cos (x))]
multiply sin (x) in both numerator and denominator
Note that sin^2 (x) + cos^2 (x) is 1
= 1 / [1 / cos (x)]
= cos (x)
