Question

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

Answer 1

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`

Answer 2

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`