Prove the following identity:
$ (1+\tan A)^2+(1+\cot A)^2=(\sec A+\operatorname{cosec} A)^2 $

Given:

$(1+tan  A)^2+(1+cot  A)^2=(sec A + cosec  A)^2$

To do:

We have to prove the given identity.

Solution:

We know that,

$tan^2A=sec^2A-1$

$cot^2A=cosec^2A-1$

$tanA=\frac{sinA}{cosA}$

$cotA=\frac{cosA}{sinA}$

LHS

$(1+\tan A)^{2}+(1+\cot A)^{2}=1+tan^2A+2tanA+1+cot^2A+2cotA$

$=1+sec^2A-1+1+cosec^2A+2cosecA+2(\frac{sinA}{cosA}+\frac{cosA}{sinA})$

$=sec^2A+cosec^2A+2(\frac{sin^2A+cos^2A}{sinAcosA})$

$=sec^2A+cosec^2A+2(\frac{1}{sinAcosA})$

$=sec^2A+cosec^2A+2secAcosecA$

RHS

$(secA+cosecA)^2=sec^2A+cosec^2A+2secAcosecA$

LHS$=$RHS

Hence proved.

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Updated on: 10-Oct-2022

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