(2cos^2x-1)(1+tan^2x)=1-tan^2x Please Help?How to prove the identity

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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First of all, we'll use the Pythagorean identity to re-write the second factor of the product:

1 + `tan^(2)` x = 1/`cos^(2)` x

The left side will become:

(2`cos^(2)`x - 1 )*(1/`cos^(2)` x)

Now, we'll remove the brackets from the left and we'll re-write the identity to be demonstrated:

2 - 1/`cos^(2)` x = 1 - `tan^(2)` x

We'll use again the Pythagorean identity to re-write the second term of the difference from the right side:

1 + `tan^(2)` x = 1/`cos^(2)` x => `tan^(2)` x = 1/`cos^(2)` x - 1

The identity to be verified will become:

2 - 1/`cos^(2)` x = 1 - (1/`cos^(2)`x - 1 )

We'll remove the brackets from the right side:

2 - 1/`cos^(2)` x = 1 - 1/`cos^(2)` x + 1

We'll combine like terms and we'll get:

2 - 1/`cos^(2)` x =  2 - 1/`cos^(2)` x

Since LHS = RHS, therefore the given identity (2`cos^(2)` x - 1)(1+`tan^(2)`x ) = 1 - `tan^(2)` x is verified.

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