Question

How do you verify the identity `sec2theta=(cos^2theta+sin^2theta)/(cos^2theta-sin^2theta)`?

Answer 1

The equation is verified by applying some trigonometric identities.
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Trigonometric identities used in this exercise:
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`color(red)(cos^2x + sin^2x =1) " " color(red)(Eq1)`
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`color(blue)(cos^2x - sin^2x = cos2x)" "color(blue)(Eq2)`
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`secx=1/ cosx" "Eq3`
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Taking the second side of the equation to prove the equality.
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`(cos^2theta + sin^2theta)/(cos^2theta-sin^2theta)`
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`=color(red)1/(cos^2theta-sin^2theta)" "`applying `color(red)(Eq1)`

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`=color(red)1/color(blue)(cos2theta)" "`applying `color(blue)(Eq2)`
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`=sec2theta" "` Applying `Eq3 " "` for `x=2theta`