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2sec^2x(1-sin^2x)

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1 Answer

Using sin2x+cos2x=1 the above expression is 2sec2x*cos2x but sec(x)=1/cos(x) so we have 2*cos2x/cos2x=2.
 
So this expression is just 2. and we used two fundamental identities to simplify.
 
Jim

Comments

Brenda,
        There are two factors 1. 2sec2x and 2. 1-sin2x let take them one at a time
 
The definition of the secant function is sec(x)=1/cos(x) now if we square both sides we have 2sec2(x)=2/cos2(x) so much for that. Now lets look at the second factor (1-sin2(x)). You should know that sin2(x)+cos2(x)=1. This is a fundamental relationship related to the Pythagorean theorem. Using this relationship we can solve it for cos2(x)=1-sin2(x)
 
If we now put both factors together we have (2/cos2(x))*(cos2(x)) the cos2(x) terms in the numerator and denominator cancel leaving the factor 2.  Q.E.D.
 
I hope this is clearer. If not let me know.
 
Jim