SOLUTION: Prove the reduction formula : csc(pi/2 - x ) = sec x

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Question 1011254: Prove the reduction formula : csc(pi/2 - x ) = sec x
Answer by mathmate(429) About Me  (Show Source):
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Question:
Prove the reduction formula : csc(pi/2 - x ) = sec x

Solution:
We will need the following identities:
csc(a)=1/sin(a).................................(1)
sin(a-b)=sin(a)cos(b)-cos(a)sin(b)..............(2)
cos(pi/2)=0.....................................(3)
sin(pi/2)=1.....................................(4)
Step 1:
reduce the left-hand side of the formula in terms of sine using identity (1)
csc(pi/2-x)=1/(sin(pi/2-x)
apply identity (2) to above:
csc(pi/2-x)
=1/(sin(pi/2-x)
=1/(sin(pi/2)cos(x)-cos(pi/2)sin(x))
=1/(1*cos(x)-0*sin(x)) ......... apply (3) and (4)
=1/cos(x)
=sec(x) ...QED