Question #3f1ae

3 Answers
1s2s2p
Jan 9, 2018

#cscx-=1/sinx#

#secx-=1/cosx#

So, now we have:
#(cosx+sinx)(1/cosx+1/sinx)#

We can now expand this to get:
#(cosx*1/cosx)+(sinx*1/sinx)+(cosx*1/sinx)+(sinx*1/cosx)#

#=(cosx/cosx)+(sinx/sinx)+(cosx/sinx)+(sinx/cosx)#

#=1+1+1/(sinx/cosx)+sinx/cosx#

#=2+1/tanx+tanx#

#=2+cotx+tanx#

sjc
Jan 9, 2018

see below

Explanation:

to prove

#(cosx+sinx)(cscx+secx)=2+tanx+cotx#

take the #LHS#

#(1) #change all the functions to either sines or cosines

#(cosx+sinx)(1/sinx+1/cosx)#

#(2)#multiply out

#cosx/sinx+cosx/cosx+sinx/sinx+sinx/cosx#

#(3)#simplify to the required result

#cotx+1+1+tanx#

giving the required result

#2+tanx+cotx#

Somebody N.
Jan 9, 2018

See below.

Explanation:

#(cos(x)+sin(x))(csc(x)+sec(x))=2+tan(x)+cot(x)#

#LHS#

Expand:

#(cos(x)+sin(x)(csc(x)+sec(x))#

#=cos(x)csc(x)+cos(x)sec(x)+sin(x)csc(x)+sin(x)sec(x)#

Identities:

#color(red)(sec(x)=1/cos(x))#

#color(red)(csc(x)=1/sin(x))#

#cos(x)csc(x)+cos(x)sec(x)+sin(x)csc(x)+sin(x)sec(x)#

#=cos(x)/sin(x)+cos(x)/cos(x)+sin(x)/sin(x)+sin(x)/cos(x)#

#=cos(x)/sin(x)+1+1+sin(x)/cos(x)#

#=cos(x)/sin(x)+2+sin(x)/cos(x)#

Identities:

#color(red)(cos(x)/sin(x)=cot(x))#

#color(red)(sin(x)/cos(x)=tan(x))#

#=cos(x)/sin(x)+2+sin(x)/cos(x)=2+tan(x)+cot(x)#

#LHS-=RHS#

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