A quick question?

Question

how do i figure this out

sinx/cscx + cosx/secx = sinx cscx

i've tried turning csc into 1/sin & sec into 1/cos but that got me to sin^2x & cos^2x to get 1. i don't kno where i went wrong but i need some help with that one!


one more i have is 2sin^2x - 1 = tanx -cotx/tanx+cotx.

i've tried switching the sin^2x to 1-cos^2x but i don't kno how i would get tan & cot.


if someone can showme how to work these problems out i'd appreciate it so much

Answer 1

First one is easy - you nearly had it; just note that
LHS = sin x / (1 / sin x) + cos x (1 / cos x) = sin^2 x + cos^2 x = 1
and RHS = sin x csc x = sin x (1 / sin x) = 1 also.

For the second one, I';d recommend starting on the RHS and trying to simplify:
RHS = (tan x - cot x) / (tan x + cot x)
= (sin x / cos x - cos x / sin x) / (sin x / cos x + cos x / sin x)
Multiply top and bottom by sin x cos x:
= (sin^2 x - cos^2 x) / (sin^2 x + cos^2 x)
= sin^2 x - cos^2 x
= sin^2 x - (1 - sin^2 x)
= 2 sin^2 x - 1
= LHS.

Answer 2

Well, you have the left side = 1; the right side is the sine function over its reciprocal, so that's also 1. Q.E.D.

(A lot of credit to Scarlet (below) for putting the grouping symbols into the right side of your second problem so that it is read correctly.)

Answer 3

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scary