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Prove the identity step by step

-tan^{2} x + sec^{2} x = 1

User Irreal
by
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1 Answer

3 votes


-tan^x + sec^2x = 1 is proved

Solution:

We have to prove that,


-tan^x + sec^2x = 1

By the trignometric identity,


sin^2x + cos^2 x = 1

Divide both the sides by
cos^2x in above identity,


(sin^2x)/(cos^2x) + (cos^2x)/(cos^2x) = (1)/(cos^2x) --- eqn 1

We know that by definition of tan,


tan x = (sinx)/(cosx)

Therefore,


tan^2x = (sin^2x)/(cos^2x)

Apply the above in eqn 1


tan^2x + 1 = (1)/(cos^2x) ---- eqn 2

By definition of cosine,


cosx = (1)/(secx)

Therefore,


cos^2x = (1)/(sec^2x)

Apply the above in eqn 2


tan^2x + 1 = sec^2x

On rewriting we get,


sec^2x - tan^2x = 1\\

Thus the given identity is proved step by step

User Michael Edmison
by
8.7k points

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