SOLUTION: Show that (a) (tanx - cosx)/sinxcosx = sec²x - cosec²x (b) sin⁴x + cos⁴x = 1- 2sin²xcos²x

Algebra ->  Trigonometry-basics -> SOLUTION: Show that (a) (tanx - cosx)/sinxcosx = sec²x - cosec²x (b) sin⁴x + cos⁴x = 1- 2sin²xcos²x      Log On


   

Question 1132184: Show that
(a) (tanx - cosx)/sinxcosx = sec²x - cosec²x
(b) sin⁴x + cos⁴x = 1- 2sin²xcos²x
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
(a)



start with

%28tan%28x%29+-+cos%28x%29%29%2F%28sin%28x%29cos%28x%29+%29

=tan%28x%29%2F%28sin%28x%29cos%28x%29+%29+-+cos%28x%29%2F%28sin%28x%29cos%28x%29+%29

=

=

=%281%2Fcos%28x%29%29%2Fcos%28x%29++-+1%2Fsin%28x%29

=1%2Fcos%5E2%28x%29++-+1%2Fsin%28x%29

=sec%5E2%28x%29+-+csc%28x%29=>this is your solution, it’s not csc%5E2%28x%29+(you probably made mistake while typing)


(b)

sin%5E4%28x%29+%2B+cos%5E4%28x%29+=+1-+2sin%5E2%28x%29cos%5E2%28x+%29

start with

sin%5E4%28x%29+%2B+cos%5E4%28x%29

=sin%5E2%28x%29+sin%5E2%28x%29+%2B+cos%5E2%28x%29cos%5E2%28x%29

=%281-cos%5E2%28x%29+%29sin%5E2%28x%29+%2B+%281-sin%5E2%28x%29%29cos%5E2%28x%29

=

=sin%5E2%28x%29%2B+cos%5E2%28x%29-2sin%5E2%28x%29%2Acos%5E2%28x%29

=1-2sin%5E2%28x%29%2Acos%5E2%28x%29




Answer by ikleyn(52832) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will solve the part  (b)  here only,  in a shorter way. 


Start with


    sin%5E2%28x%29 + cos%5E2%28x%29 = 1.


Square both sides


    sin%5E4%28x%29 + 2%2Asin%5E2%28x%29%2Acos%5E2%28x%29 + cos%5E4%28x%29 = 1


Move the term  2%2Asin%5E2%28x%29%2Acos%5E2%28x%29  from the left side to the right, changing the sign


    sin%5E4%28x%29 + cos%5E4%28x%29 = 1 - 2%2Asin%5E2%28x%29%2Acos%5E2%28x%29.

QED.