SOLUTION: write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression sin^2x + sin^2 x cot^2 x and sin^2x-1/cos(-x)

Algebra ->  Trigonometry-basics -> SOLUTION: write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression sin^2x + sin^2 x cot^2 x and sin^2x-1/cos(-x)      Log On


   

Question 852861: write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression
sin^2x + sin^2 x cot^2 x
and
sin^2x-1/cos(-x)
Answer by DrBeeee(684) About Me  (Show Source):
You can put this solution on YOUR website!
Given;
(1) sin%5E2%28x%29+%2B+sin%5E2%28x%29%2Acot%5E2%28x%29
Use the identity
(2) cot%28x%29+=+cos%28x%29%2Fsin%28x%29
and square both sides to get
(3) cot%5E2%28x%29+=+cos%5E2%28x%29%2Fsin%5E2%28x%29
Now substitute (3) into (1) and get
(4)sin%5E2%28x%29+%2B+sin%5E2%28x%29%2Acos%5E2%28x%29%2Fsin%5E2%28x%29 or
(5)sin%5E2%28x%29+%2B+cos%5E2%28x%29
Since
(6) sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1
You have
(7)sin%5E2%28x%29+%2B+sin%5E2%28x%29%2Acot%5E2%28x%29+=+1
Answer: 1
Also given
(8) %28sin%5E2%28x%29+-+1%29%2Fcos%28-x%29
Using (6) we get
(9) sin%5E2%28x%29+=+1+-+cos%5E2%28x%29+
Now put (9) into (8) to get
(10) %28+1+-+cos%5E2%28x%29+-+1%29%2Fcos%28-x%29 or
(11)%28-cos%5E2%28x%29%29%2Fcos%28-x%29 and
Since the cosine is an even function we have
(12) +cos%28-x%29+=+cos%28x%29
Then (11) reduces to
(13) -cos%28x%29
Answer: -cos%28x%29