Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 years ago

# Help with trig identities?

Im stuck on this one..

(secA+tanA)/(secA-tanA) - (1-sinA)/(1+sinA)

=

4tanAcscA

??

Relevance

OK, here you go sweetheart :)

Note the identities used in the proof :

secA = 1/cosA

tanA = sinA/cosA

1 - sin²A = cos²A

Starting with the LHS we have :

∴ LHS

= [(secA + tanA) /(secA - tanA)] - [(1 - sinA) /(1 + sinA)

= [(1/cosA + sinA/cosA) /(1/cosA - sinA/cosA)] - [(1 - sinA) /(1 + sinA)]

= [[(1 + sinA) /cosA] ÷ [(1 - sinA) /cosA]] - [(1 - sinA) /(1 + sinA)]

= [(1 + sinA) /(1 - sinA)] - [(1 - sinA) /(1 + sinA)]

= [(1 + sinA)² - (1 - sinA)²] /(1 - sin²A)

= [1 + 2sinA + sin²A - 1 + 2sinA - sin²A] /(1 - sin²A)

= [4sinA] /cos²A

= 4 (sinA/cosA) (1/cosA)

= 4(tanA)(secA)

= RHS, as required, Q.E.D.

Oh, I forgot to mention that what you've written in your question is NOT an identity! It should be a 'secA' instead of a 'cosecA' at the end for the identity to be valid.

Hope this helps !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

P.S. (Don't forget to vote me best answer as being the first to correctly and thoroughly answer your question!)

• TomV
Lv 7
2 years agoReport

You didn't really answer the question, 'though. The RHS of the posted identity is 4tanAcscA, not 4 tanAsecA.

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• Take it or leave it, but on these identity proofs I always begin by checking whether there are any conflicts in the domains of the two sides of the equation. In this case there are.

Let A = 0.

The left side of the equation is zero. The right side is undefined. When a real number is equated with an undefined expression, the equation cannot be an identity. And no, there is no such thing as expressions that are almost an identical.

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• The reason you are stuck on that one may be because it is not an identity.

Let A = 60°

sinA = √3/2

secA = 1/cosA = 2

cscA = 1/sinA = 2/√3

tanA = √3

(secA+tanA)/(secA-tanA) - (1-sinA)/(1+sinA) = 4tanAcscA

(2 + √3)/(2 - √3) - (1-√3/2)/(1+√3/2) = 4(√3)(2/√3)

(2+√3)/(2-√3) - (2-√3)/(2+√3) = 8

(2+√3)²/(4-3) - (2-√3)²/(4-3) = 8

4 + 4√3 + 3 - (4 - 4√3 + 3) = 8

7 + 4√3 - 7 + 4√3 = 8

8√3 = 8 : False statement

Equation is not an identity.

However

(secA+tanA)/(secA-tanA) - (1-sinA)/(1+sinA) = 4tanAsecA

is an identity. (Note the different RHS)

LHS = (secA+tanA)/(secA-tanA) - (1-sinA)/(1+sinA)

replace secA with 1/cosA and tanA with sinA/cosA

= (1/cosA + sinA/cosA)/(1/cosA - sinA/cosA) - (1-sinA)/(1+sinA)

multiply numerator and denominator of first term by cosA

= (1+sinA)/(1-sinA) - (1-sinA)/(1+sinA)

multiply numerator and denominator of first term by (1+sinA) and second term by (1-sinA)

= (1+sinA)²/(1-sin²A) - (1-sinA)²/(1 - sin²A)

replace 1 - sin²A with cos²A per Pythagorean Identity, and combine terms with common denominator

= [(1+sinA)² - (1-sinA)²]/cos²A

expand squared terms

= [(1 + 2sinA + sin²A - (1 - 2sinA + sin²A)]/cos²A

combine like terms

= 4sinA/cos²A

replace sinA/cosA with tanA and 1/cosA with secA

= 4tanAsecA

= RHS (as modified)

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• Try multiplying the fractions by their conjugate (sec A + tan A) or (sec A - tan A) and (1 - sin A) or (1 + sin A) to create identities (sec²A - tan²A = 1) and (1 - sin²A = cos²A)

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