Prove the following identities (1-17)
Question:

Prove the following identities (1-17)

$(1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2}=\sec ^{2} \alpha \sec ^{2} \beta$

Solution:

$(1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2}=\sec ^{2} \alpha \sec ^{2} \beta$

$\mathrm{LHS}=(1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2}$

$=1+\tan ^{2} \alpha \tan ^{2} \beta+2 \tan \alpha \tan \beta+\tan ^{2} \alpha+\tan ^{2} \beta-2 \tan \alpha \tan \beta$

$=1+\tan ^{2} \alpha \tan ^{2} \beta+\tan ^{2} \alpha+\tan ^{2} \beta$

$=\tan ^{2} \alpha\left(\tan ^{2} \beta+1\right)+1\left(1+\tan ^{2} \beta\right)$

$=\left(1+\tan ^{2} \beta\right)\left(1+\tan ^{2} \alpha\right)$

$=\sec ^{2} \alpha \cdot \sec ^{2} \beta$

$=\mathrm{RHS}$

Hence proved.

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