How do you simplify #(2-sec^2x)/(sec^2x)#?

Question

Daniel Acosta · Accepted Answer

#(2-sec^2(x))/sec^2(x) = cos(2x)#

First, split the fraction.

#(2 - sec^2(x))/sec^2(x)#

#= 2/sec^2(x) - sec^2x/sec^2(x)#

Because #cos(x) = 1/sec(x)#, #color(red)(cos^2(x) = 1/sec^2(x))#

#= 2/color(red)(sec^2(x)) - sec^2(x)/sec^2(x)#

#= color(blue)(2cos^2(x) - 1)#

Observe the following:

#color(blue)cos(2x)#

#= cos^2(x) - sin^2(x)#

# = cos^2(x) - (1 - cos^2x)#

# = color(blue)(2cos^2(x) - 1)#

Therefore,

#(2-sec^2(x))/sec^2(x) = cos(2x)#

Emery Adkins · Answer

It simplifies to #cos(2x)#.

Use these identites: #cos(2x)=2cos^2x-1# #secx=1/cosxqquadcolor(blue)=>qquadsec^2x=1/cos^2x# First, split the fraction: #color(white)=(2-sec^2x)/sec^2x# #=2/sec^2x-sec^2x/sec^2x# #=2/sec^2x-1# #=2*1/sec^2x-1# #=2*1/(1/cos^2x)-1# #=2*cos^2x-1# #=2cos^2x-1# #=cos(2x)# Hope this helped!

Grace Ashcraft · Answer

undefined To simplify the expression (2 - sec^2x) / sec^2x, you can start by factoring out a common factor from the numerator:

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

Then, rewrite the original expression with the factored form:

(2cos^2x - 1)/cos^2x / sec^2x

Next, remember that sec^2x is equal to 1/cos^2x. So, you can rewrite sec^2x as 1/cos^2x in the denominator:

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

Now, when you divide by a fraction, you can multiply by its reciprocal. So, rewrite the expression as multiplication by the reciprocal of the denominator:

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

Now, you can cancel out the common terms:

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

So, the simplified expression is 2cos^2x - 1.