Prove that #tanx/(1+secx)+(1+secx)/tanx=2secx#?

2 Answers
Feb 23, 2016

It is not an identity. It cannot be verified.

Explanation:

The left side is not always equal to the right. For example at #x=pi/6# the left side is #4# and the right side is #(4sqrt3)/3#

Feb 23, 2016

It should be #tanx/(1+secx)+(1+secx)/tanx=2cscx#. See proof below.

Explanation:

#tanx/(1+secx)+(1+secx)/tanx!=2secx#. It should rather be #tanx/(1+secx)+(1+secx)/tanx=2cscx#

Perhaps what you mean is to prove that #tanx/(1+secx)+(1+secx)/tanx=2cscx#. To solve this let us start from LHS and prove using other identities that this is equal to RHS.

#tanx/(1+secx)+(1+secx)/tanx# (let us first add them like fractions)

= #(tan^2x+(1+secx)^2)/(tanx(1+secx)# (expanding this)

= #(tan^2x+1+sec^2x+2secx)/(tanx(1+secx)#(using #sec^2x=1+tan^2x#)

= #(sec^2x+sec^2x+2secx)/(tanx(1+secx)# or

=#(2sec^2x+2secx)/(tanx(1+secx)# or

= #(2secx(secx+1))/(tanx(1+secx)#

Now, using #secx=1/cosx# and #tanx=sinx/cosx#

= #2secx/tanx# =#(2/cosx)/(sinx/cosx)# (simplifying & using #1/sinx=cscx#)

= #(2/sinx)# = #2cscx#