secx - sinxtanx
1/cosx - sinx/cosx = (1-sinx)/cosx
I guess you could go further, recalling that sin and cos are complementary functions, and get
(1-sinx)/cosx
= (1-cos(90-x))/sin(90-x)
= tan((90-x)/2)
= tan(45 - x/2)
Oops -- my bad!
secx - sinx tanx
= 1/cosx - sin^2x/cosx
= (1-sin^2x)/cosx
= cos^2x/cosx
= cosx
To simplify the expression sec(x) - sin(x)tan(x), we can use some trigonometric identities to rewrite it in a simpler form.
Step 1: Recall the definitions of sec(x) and tan(x):
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)
Step 2: Substitute the definitions of sec(x) and tan(x) into the expression:
sec(x) - sin(x)tan(x) = 1/cos(x) - sin(x) * (sin(x)/cos(x))
Step 3: Simplify the expression using the common denominator cos(x):
= 1/cos(x) - sin^2(x)/cos(x)
Step 4: Combine the fractions by finding a common denominator:
= (1 - sin^2(x))/cos(x)
Step 5: Apply the Pythagorean Identity sin^2(x) + cos^2(x) = 1:
= cos^2(x)/cos(x)
Step 6: Simplify and cancel out the cos(x) terms:
= cos(x)
Therefore, the simplified expression is cos(x).
To simplify the expression sec(x) - sin(x)tan(x), we can use trigonometric identities.
First, let's simplify the expression sin(x)tan(x):
tan(x) can be expressed as sin(x)/cos(x). So, sin(x)tan(x) = sin(x) * (sin(x)/cos(x)) = (sin^2(x))/cos(x).
Now, the expression becomes sec(x) - (sin^2(x))/cos(x).
Next, let's simplify the expression sec(x):
Using the reciprocal identity, sec(x) = 1/cos(x).
So, the expression now becomes 1/cos(x) - (sin^2(x))/cos(x).
To combine these fractions, multiply the first fraction by cos(x)/cos(x) and the second fraction by 1/1:
(1 * cos(x))/(cos(x) * cos(x)) - (sin^2(x))/cos(x).
Simplifying further:
= cos(x)/cos^2(x) - (sin^2(x))/cos(x).
Now, let's simplify the numerator:
cos(x)/cos^2(x) = 1/cos(x).
Finally, we have:
1/cos(x) - (sin^2(x))/cos(x) = (1 - sin^2(x))/cos(x).
The simplified expression is (1 - sin^2(x))/cos(x), which can be further simplified to cos^2(x)/cos(x) = cos(x).