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Prove that 1 - sec x/1 + sec x is equal to cos x - 1/cos x + 1

Prove that 1 - sec x/1 + sec x is equal to cos x - 1/cos x + 1-example-1
User Ethanneff
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2 Answers

3 votes

Answer: See below

Explanation:


(1-\sec x)/(1+\sec x)\quad =\quad (\cos x-1)/(\cos x+1)

LHS --> RHS

Convert sec x to 1/(cos x):


(1-(1)/(\cos x))/(1+(1)/(\cos x))

Find a common denominator for the numerator and denominator:


((\cos x)/(\cos x)-(1)/(\cos x))/((\cos x)/(\cos x)+(1)/(\cos x))\quad = \quad ((\cos x-1)/(\cos x))/((\cos x+1)/(\cos x))

Eliminate the denominators:


\quad (\cos x-1)/(\cos x+1)

LHS = RHS so identity is proven.

User Mohamed Thoufeeque
by
7.9k points
6 votes

Answer:


(1-secx)/(1+secx) =
(cosx - 1)/(cosx + 1)

Explanation:

We are to prove that 1 - sec x/1 + sec x is equal to cos x - 1/cos x + 1

To do this, we need to know that secx= 1/cosx

from the question given (1-secx) ÷ (1+secx)

Lets first simplify (1-secx)

1-secx = 1 - 1/cosx =
(cosx - 1)/(cosx)

Also we will go ahead and simplify (1+secx)

1 + secx = 1 + 1/cosx =
(cosx + 1)/(cosx)


(1-secx)/(1+secx) =
(cosx - 1)/(cosx) ÷
(cosx + 1)/(cosx)

=
(cosx - 1)/(cosx) ×
(cosx)/(cosx + 1)

(The cosx at the numerator will cancel-out the cosx at the denominator)

=
(cosx - 1)/(cosx + 1)


(1-secx)/(1+secx) =
(cosx - 1)/(cosx + 1)

Hence proved.

User Mario Camou
by
8.5k points

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