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# Prove that (1-tanx)(1-cotx)=2-secxcscx?

I can usually get these but i'm really lost on this. Thanks for the help

### 6 Answers

- RyomaLv 710 years agoFavorite Answer
Hi

Recall that:

cot(x) = 1/tan(x), 1/sin(x) = csc(x), 1/cos(x) = sec(x)………(1) (reciprocal identities)

tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x)………(2) (quotient identities)

cos^2(x) + sin^2(x) = 1………(3) (pythagorean identity)

[1 - tan(x)]*[1 - cot(x)]

= 1*1 + 1*-cot(x) - tan(x)*1 - tan(x)*-cot(x) <--- by expanding it out

= 1 - cot(x) - tan(x) + tan(x)cot(x)

= 1 - cot(x) - tan(x) + tan(x)/tan(x) <--- by (1)

= 1 - cot(x) - tan(x) + 1

= 2 - cot(x) - tan(x)

= 2 - cos(x)/sin(x) - sin(x)/cos(x) <--- by (2)

= 2 - cos^2(x)/[sin(x)cos(x)] - sin^2(x)/[sin(x)cos(x)] <--- common denominator

= 2 - [cos^2(x) + sin^2(x)]/[sin(x)cos(x)]

= 2 - 1/[sin(x)cos(x)] <--- by (3)

= 2 - [1/sin(x)]*[1/cos(x)]

= 2 - csc(x)*sec(x) <--- by (2)

= 2 - sec(x)csc(x)

I hope this helps!

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- 4 years ago
scythian is on the mark here: the positive integers are rigorously constructed using Peano's axioms, and 1+1 is the immediate successor of 1, which is defined as 2. But I would wholly disagree with scythian's argument that Godel's Incompleteness Theorems are controversial; the only mathematicians that seriously feel this way are fringe logicians and maniacal set theorists. Godel's theorems opened up alot of new venues in mathematics, and some might say "freed" us from the almost certain rigorous demise to which we were headed. Basically, Hilbert in 1900 asked "could somebody please set up a system of axioms which is completely consistent and serves as a basis for all math?", to which Godel responded, several years later, "no, no one can; any axiomatic system describing the integers will have certain unprovable statements, and some which are consistent when treated both as true and as false." To summarize: 1+1 is 2 because it is defined that way, axiomatically, and thus cannot be proven under the standard system of Peano's axioms. Steve EDIT - Above, when I say "unprovable statements", I mean statements treated as true, but not proved as such (not including axioms). When I talk about a statement being "consistent when treated both as true and as false", I mean independent of the current axiomatic framework; this is equivalent to saying the framework cannot prove its own consistency.

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- svLv 710 years ago
(1-tanx)(1-cotx)

= 1 -- tanx -- cotx + 1

= 2 -- [sinx / cosx + cosx / sinx]

= 2 -- (sin^2x + cos^2x) / sinxcosx

= 2 -- 1/sinxcosx

= 2 -- cosecxsecx

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- MechEng2030Lv 710 years ago
Foiling the L.H.S:

1 - 1/tan(x) - tan(x) + 1

2 - 1/tan(x) - tan(x)

2 - (1/tan(x) + tan(x))

2 - (1 + tan²(x))/(tan(x))

2 - (1/cos²(x)) * cos(x)/sin(x)

2 - sec(x)*csc(x)

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