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natima [27]
3 years ago
8

(Partially Done!) Verify the identity and show your work.

Mathematics
1 answer:
vladimir1956 [14]3 years ago
7 0
Throughout all of these steps I'm only going to alter the left hand side (LHS). I am NOT going to change the right hand side (RHS) at all.

Before I change the LHS of the original equation, let's focus on the given identity
cot^2(x) + 1 = csc^2(x)

Since we know it's an identity, we can subtract 1 from both sides and the identity would still hold true
cot^2(x) + 1 = csc^2(x)
cot^2(x) + 1-1 = csc^2(x)-1
cot^2(x) + 0 = csc^2(x)-1
cot^2(x) = csc^2(x)-1

So we'll use the identity cot^2(x) = csc^2(x)-1

---------------------------------------------

Now onto the main equation given

cot^2(x) + csc^2(x) = 2csc^2(x) - 1
cot^2(x) + csc^2(x) = 2csc^2(x) - 1 .... note the term in bold
csc^2(x)-1 + csc^2(x) = 2csc^2(x) - 1 .... note the terms in bold
[ csc^2(x) + csc^2(x) ] - 1 = 2csc^2(x) - 1
[ 2csc^2(x) ] - 1 = 2csc^2(x) - 1
2csc^2(x) - 1 = 2csc^2(x) - 1

The bold terms indicate how the replacements occur.

So the original equation has been proven to be an identity because the LHS has been altered to transform into the RHS
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