How do you find a double angle formula for sec(2x) in terms of only csc(x) and sec(x)?

2 Answers

See Below.

Explanation:

sec(2x)

= 1/cos (2x)

= 1/(cos (x + x))

= 1/(cos x * cos x + sin x * sin x) [Expanded using addition identity]

= 1/((1/sec x) * (1/secx) + (1/csc x) * (1/csc x))

= 1/((csc^2x + sec^2x)/(sec^2x * sin^2 x)) [Simple Addition]

= (sec^2 x csc^2x)/(csc^2x + sec^2 x)

Hope this helps.

color(blue)(=>(sec^2x * csc^2x) / (csc^2x - sec^2x)

Explanation:

sec 2x = 1 / (cos 2x)

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=> 1 / (cos^2 x - sin^2x) using double angle formula

=> 1 / (1/sec^2x - 1/csc^2x)

=> 1 /( (csc^2x - sec^2x) / (sec^2x csc@^2x))

=>(sec^2x * csc^2x) / (csc^2x - sec^2x)