Which of the following trigonometric expressions is equal to sec 6 θ ? Options: A . &nbsptan6 θ+3 tan2 θ sec2 θ+1 B . &nbsptan6 θ−3 tan2 θ sec2 θ+1 C . &nbsptan6 θ−1 D . &nbsptan6 θ+1

Which of the following trigonometric expressions is equal to sec6 θ ?

Question

Which of the following trigonometric expressions is equal to

{sec}^{6} θ

Options:

A . tan6 θ+3 tan2 θ sec2 θ+1

B . tan6 θ−3 tan2 θ sec2 θ+1

C . tan6 θ−1

D . tan6 θ+1

Answer: Option A
:
A

{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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