How do you prove # sec^2xcscx =sec^2x+csc^2x #?

2 Answers
Apr 21, 2018

Check the Problem.

Explanation:

I have doubt about the validity of the ("so called") Identity.

Had it been an Identity, it would have hold good for #x=pi/4#.

#"For "x=pi/4, "The L.H.S."=sec^2(pi/4)csc(pi/4)#

#=(sqrt2)^2sqrt2=2sqrt2#, whereas,

#"The R.H.S.=2+2=4"#.

#:."The L.H.S."!="The R.H.S."#

In fact, #sec^2xcsc^2x=sec^2x+csc^2x#.

#sec^2x+csc^2x=1/cos^2x+1/sin^2x#,

#=((sin^2x+cos^2x))/(cos^2xsin^2x)#,

#=(1)-:{1/sec^2x*1/csc^2x}#,

#=sec^2xcsc^2x#.

Apr 21, 2018

#sec^2xcscx!=sec^2x+csc^2x,...why ?#
Please see below.

Explanation:

We know that,

#color(blue)((1)sectheta=1/costheta and csctheta=1/sintheta#

#color(red)((2)sin^2theta+cos^2theta=1#

Here,

#LHS=color(violet)(sec^2xcscx) and RHS=sec^2x+csc^2x#

We take ,

#RHS=color(blue)(sec^2x+csc^2x#

#=color(blue)(1/cos^2x+1/sin^2x...toApply(1)#

#=color(red)((sin^2x+cos^2x))/(cos^2xsin^2x)#

#=color(red)(1)/(cos^2xsin^2x)...toApply(2)#

#=1/cos^2x*1/sin^2x#

#=color(violet)(sec^2xcsc^2x#

#!=LHS#

Hence, #sec^2xcolor(red)(cscx!=)sec^2x+csc^2x#,

But, #sec^2xcolor(red)(csc^2x=)sec^2x+csc^2x#