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Trigonometric Identities

Which of the ...

Question

Which of the following trigonometric expressions is equal to ${sec}^{6} θ$ ?

A

${tan}^{6} θ + 3 {tan}^{2} θ {sec}^{2} θ + 1$

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B

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

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C

${tan}^{6} θ - 1$

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D

${tan}^{6} θ + 1$

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Solution

The correct option is A
${tan}^{6} θ + 3 {tan}^{2} θ {sec}^{2} θ + 1$

${sec}^{6} θ = ({sec}^{2} θ)^{3}$

$= ({tan}^{2} θ + 1)^{3} \dots \dots (1 + {tan}^{2} θ = {sec}^{2} θ)$

$= {tan}^{6} θ + 3 {tan}^{4} θ + 3 {tan}^{2} θ + 1$

$= {tan}^{6} θ + 3 {tan}^{2} θ ({tan}^{2} θ) + 3 {tan}^{2} θ + 1$

$= {tan}^{6} θ + 3 {tan}^{2} θ ({sec}^{2} θ - 1) + 3 {tan}^{2} θ + 1$

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

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

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