1+tan^2A +(1+1upon tan^2A) = 1upon sin^2A-sin^4A . proof this and how it is sin^4A explain it in detail

Respected User,

It is given that 1 + tan ² A + (1 + 1/tan ² A)

and we have to prove that 1 + tan ² A + (1 + 1/tan ² A)  = ______1______

(sin ² A – sin ⁴ A)

So, let

1 + tan ² A + (1 + 1/tan ² A) _____ eqn. 1

Now, since 1 + tan ² x = sec ² x

Therefore, eqn. 1 can be written as

sec ² A + ((tan ² A+1)/tan ² A)

which will further reduce to

sec ² A + (sec ² A/tan ² A)

= sec ² A + (cos ² A/cos ² A*sin ² A)

= sec ² A + (1/sin ² A)

Again, secA = 1/cosA

Thus, rewriting sec ² A as 1/cos ² A

= 1/cos ² A + 1/sin ² A

Now, take LCM

= sin ² A + cos ² A/(sin ² A*cos ² A)

Rewrite sin ² A + cos ² A = 1 and cos ² A as 1 – sin ² A

= 1/ (sin ² A*(1 – sin ² A))

Now, multiply the values in denominator.

= 1/ (sin ² A*1 – (sin ² A)(sin ² A))

= 1/(sin ² A – sin ⁴ A)

Hope, I was able to solve your query.