What is a simplified form of the expression (sec^3 theta) - (sec theta/cot^2 theta)?
a. 0
b. sec theta-tan theta
c. cos theta
d. sec theta*****
Every time I attempt to work out this problem I end up with one. I know the answer is sec theta, but I do not know how to get there. Can someone show me how to get to this?
secT(sec^2T-1/ctn^T)
secT(1/cos^2T-sin^2T/cos^2T)
secT(1-sin^2T)/cos^2T
secT(cos^2T)/cos^2T
secT
Which is a simplified form of the expression -4(3r – 2) – 2(r + 1)?
What is ctn^T in the first step?
ctn= cotangent
sec 0 (theta) is the correct answer
Yes, that's correct! Well done!
To simplify the given expression, let's break it down step by step:
Expression: (sec^3 theta) - (sec theta/cot^2 theta)
Step 1: Rewrite sec^3 theta as (sec theta)^3
Expression: (sec theta)^3 - (sec theta/cot^2 theta)
Step 2: Rewrite cot^2 theta as (cos theta/sin theta)^2
Expression: (sec theta)^3 - (sec theta / (cos^2 theta/sin^2 theta))
Step 3: Simplify the expression within the parentheses
Expression: (sec theta)^3 - (sec theta * sin^2 theta / cos^2 theta)
Step 4: Simplify the right side of the expression by multiplying through by cos^2 theta
Expression: (sec theta)^3 - (sec theta * sin^2 theta / cos theta)
Step 5: Simplify the right side of the expression by multiplying through by sec theta
Expression: (sec theta)^3 - (sec theta * sin^2 theta * sec theta / cos theta)
Step 6: Simplify the right side of the expression by multiplying through by sec theta
Expression: (sec theta)^3 - (sin^2 theta * sec^2 theta / cos theta)
Step 7: Recall the trigonometric identity: sin^2 theta + cos^2 theta = 1.
Substitute 1 - cos^2 theta for sin^2 theta
Expression: (sec theta)^3 - ((1 - cos^2 theta) * sec^2 theta / cos theta)
Step 8: Distribute the denominator (1 - cos^2 theta) through
Expression: (sec theta)^3 - (sec^2 theta - cos^2 theta * sec^2 theta / cos theta)
Step 9: Combine like terms within the parentheses
Expression: (sec theta)^3 - (sec^2 theta - sec^2 theta * cos^2 theta / cos theta)
Step 10: Simplify further by factoring out sec^2 theta from the second term
Expression: (sec theta)^3 - (sec^2 theta * (1 - cos^2 theta) / cos theta)
Step 11: Substitute the trigonometric identity: cos^2 theta = 1 - sin^2 theta
Expression: (sec theta)^3 - (sec^2 theta * (1 - (1 - sin^2 theta)) / cos theta)
Step 12: Simplify the expression within the parentheses
Expression: (sec theta)^3 - (sec^2 theta * sin^2 theta / cos theta)
Step 13: Simplify further by canceling out sec^2 theta / cos theta
Expression: (sec theta)^3 - (sin^2 theta * sec theta)
Step 14: Recall the trigonometric identity: tan theta = sin theta / cos theta
Substitute sin theta / cos theta for sec theta
Expression: (tan theta)^3 - (tan^2 theta * sin theta)
Step 15: Recall the trigonometric identity: tan^2 theta = sec^2 theta - 1
Substitute (sec^2 theta - 1) for tan^2 theta
Expression: (tan theta)^3 - ((sec^2 theta - 1) * sin theta)
Step 16: Simplify the expression within the parentheses
Expression: (tan theta)^3 - (sec^2 theta * sin theta) + sin theta
So, the simplified form of the expression (sec^3 theta) - (sec theta/cot^2 theta) is (tan theta)^3 - (sec^2 theta * sin theta) + sin theta.
Please note that the correct option from the given answer choices is d. sec theta.