[Note: i did these wrong apparently.. :( ]
Simplify the trigonometry expressions as much as possible.
(a). tan^(2)xsecx+secx
(b).(csct-sint)/(csct)
tan^(2)xsecx+secx
= secx( tan^2 x + 1) , common factor
= secx(sec^2 x), by identity
= sec^3 x
(csct-sint)/(csct)
= (1/sint - sint)/(1/sint)
multiply top and bottom by sint
= (1 - sin^2 t)/1
= cos^2 t
Thank you so much!
No problem! I can help you with those trigonometry expressions.
(a) To simplify the expression tan^2(x)sec(x) + sec(x), we can use the trigonometric identity: sec(x) = 1/cos(x).
So, applying the identity, the expression becomes:
tan^2(x) / cos(x) + 1 / cos(x)
Now, we need to find a common denominator, which is cos(x). For the first term, tan^2(x) / cos(x), we can write it as (sin^2(x) / cos^2(x)) / cos(x). This simplifies to sin^2(x) / cos^3(x).
Combining the terms, the expression becomes:
(sin^2(x) + 1) / cos^3(x)
This is the simplified form of the expression tan^2(x)sec(x) + sec(x).
(b) Let's simplify the expression (csc(x) - sin(x)) / csc(x).
To simplify this expression, we will first find a common denominator, which is csc(x). Rewriting the expression, we have:
[(csc(x) - sin(x)) / 1] / (csc(x) / 1)
Next, we can apply the division rule to simplify by multiplying the numerator by the reciprocal of the denominator:
[(csc(x) - sin(x)) / 1] * (1 / csc(x))
This simplifies to:
(csc(x) - sin(x)) / csc(x)
Now, we can simplify further by using the trigonometric identity: csc(x) = 1/sin(x). Applying the identity, the expression becomes:
(1/sin(x) - sin(x)) / (1/sin(x))
Expanding the numerator, we have:
(1 - sin^2(x)) / sin(x)
Using the Pythagorean identity: sin^2(x) + cos^2(x) = 1, we can substitute sin^2(x) with 1 - cos^2(x). This gives us:
(1 - (1 - cos^2(x))) / sin(x)
Simplifying further, we have:
cos^2(x) / sin(x)
This is the simplified form of the expression (csc(x) - sin(x)) / csc(x).