How do you verify the identity $3 {sec}^{2} θ {tan}^{2} θ + 1 = {sec}^{6} θ - {tan}^{6} θ$ $3sec^2thetatan^2theta+1=sec^6theta-tan^6theta$ ?

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How do you verify the identity $3 {sec}^{2} θ {tan}^{2} θ + 1 = {sec}^{6} θ - {tan}^{6} θ$ $3sec^2thetatan^2theta+1=sec^6theta-tan^6theta$ ?

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Answer 1 · 2016-10-03T18:23:37

See below

Explanation:

$3 {sec}^{2} θ {tan}^{2} θ + 1 = {sec}^{6} θ - {tan}^{6} θ$ $3sec^2thetatan^2theta+1=sec^6theta-tan^6theta$

Right Side $= {sec}^{6} θ - {tan}^{6} θ$ $=sec^6theta-tan^6theta$
$= {({sec}^{2} θ)}^{3} - {({tan}^{2} θ)}^{3}$ $=(sec^2theta)^3-(tan^2theta)^3$ ->use difference of two cubes formula

$= ({sec}^{2} θ - {tan}^{2} θ) ({sec}^{4} θ + {sec}^{2} θ {tan}^{2} θ + {tan}^{4} θ)$ $=(sec^2theta-tan^2theta) (sec^4theta+sec^2thetatan^2theta+tan^4theta)$

$= 1 \cdot ({sec}^{4} θ + {sec}^{2} θ {tan}^{2} θ + {tan}^{4} θ)$ $=1 *(sec^4theta+sec^2thetatan^2theta+tan^4theta)$

$= {sec}^{4} θ + {sec}^{2} θ {tan}^{2} θ + {tan}^{4} θ$ $=sec^4theta+sec^2thetatan^2theta+tan^4theta$

$= {sec}^{2} θ {sec}^{2} θ + {sec}^{2} θ {tan}^{2} θ + {tan}^{2} θ {tan}^{2} θ$ $=sec^2theta sec^2 theta+sec^2thetatan^2theta+tan^2theta tan^2 theta$

$= {sec}^{2} θ ({tan}^{2} θ + 1) + {sec}^{2} θ {tan}^{2} θ + {tan}^{2} θ ({sec}^{2} θ - 1)$ $=sec^2theta(tan^2theta+1) +sec^2thetatan^2theta+tan^2theta(sec^2theta-1)$

$= {sec}^{2} θ {tan}^{2} θ + {sec}^{2} θ + {sec}^{2} θ {tan}^{2} θ + {sec}^{2} θ {tan}^{2} θ - {tan}^{2} θ$ $=sec^2thetatan^2theta+sec^2theta+sec^2thetatan^2theta+sec^2thetatan^2theta-tan^2theta$

$= {sec}^{2} θ {tan}^{2} θ + {sec}^{2} θ {tan}^{2} θ + {sec}^{2} θ {tan}^{2} θ + {sec}^{2} θ - {tan}^{2} θ$ $=sec^2thetatan^2theta+sec^2thetatan^2theta+sec^2thetatan^2theta+sec^2theta-tan^2theta$

$= 3 {sec}^{2} θ {tan}^{2} θ + 1$ $=3sec^2thetatan^2theta +1$

$=$ $=$ Left Side

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How do you verify the identity $3 {sec}^{2} θ {tan}^{2} θ + 1 = {sec}^{6} θ - {tan}^{6} θ$ $3sec^2thetatan^2theta+1=sec^6theta-tan^6theta$ ?

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How do you verify the identity 3sec2θtan2θ+1=sec6θ−tan6θ3sec^2thetatan^2theta+1=sec^6theta-tan^6theta?

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How do you verify the identity $3 {sec}^{2} θ {tan}^{2} θ + 1 = {sec}^{6} θ - {tan}^{6} θ$ $3sec^2thetatan^2theta+1=sec^6theta-tan^6theta$ ?