SOLUTION: -1/(csc(-theta)) - (cot(-theta))/(sec(-theta)) = csc(theta)

Algebra ->  Trigonometry-basics -> SOLUTION: -1/(csc(-theta)) - (cot(-theta))/(sec(-theta)) = csc(theta)      Log On


   

Question 1153470: -1/(csc(-theta)) - (cot(-theta))/(sec(-theta)) = csc(theta)
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
manipulate left side
-1%2Fcsc%28-theta%29+-+%28cot%28-theta%29%2Fsec%28-theta%29%29.........since csc%28-theta%29+=-csc%28theta%29,cot%28-theta%29=-cot%28theta%29, and sec%28-theta%29=sec%28theta%29....common denominator is -csc%28theta%29sec%28theta%29,

=-1%2F%28-csc%28theta%29%29+-+%28-cot%28theta%29%2Fsec%28theta%29%29

=1%2Fcsc%28theta%29%2Bcot%28theta%29%2Fsec%28theta%29

=

=%28sec%28theta%29%2Bcot%28theta%29csc%28theta%29%29%2F%28csc%28theta%29sec%28theta%29%29

express all with sin%28theta%29+and cos%28theta%29:
sec%28theta%29=1%2Fcos%28theta%29
csc%28theta%29=1%2Fsin%28theta%29+
cot%28theta%29=cos%28theta%29%2Fsin%28theta%29+


=

=

=

=....sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1

=

=%28sin%28theta%29%2Acos%28theta%29%29%2F%28cos%28theta%29%2Asin%5E2%28theta%29%29...simplify

=

=1%2Fsin%28theta%29........use identity csc%28theta%29=1%2Fsin%28theta%29+

=csc%28theta%29

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Simplify the first term on the left by expressing everything in sin(x) and cos(x), and using sin(-x) = -sin(x):

-1%2Fcsc%28-x%29+=+-sin%28-x%29+=+-%28-sin%28x%29%29+=+sin%28x%29

Similarly, simplify the second term on the left:



The expression on the left is now



Combine terms using a common denominator and use sin^2(x)+cos^2(x)=1:

%28sin%5E2%28x%29%2Bcos%5E2%28x%29%29%2Fsin%28x%29+=+1%2Fsin%28x%29+=+csc%28x%29