Prove the identity. tan^2x/secx=secx-cosx

Rui Baldwin

Rui Baldwin

Answered question

2020-12-29

Prove the identity. tan2xsecx=secxcosx

Answer & Explanation

Sally Cresswell

Sally Cresswell

Skilled2020-12-30Added 91 answers

The first thing that comes to my mind is I want everything in terms of sin's and cos's if I can, so let's multiply through by cosx so that we can at least get rid of that secx=1cosx term.
(tan2xsecx)cosx=1cos2x
Now that 1cos2x is popping out at me because I remember the Pythagorean identity sin2x+cos2x=1
Using this we see that 1cos2x=sin2x and so what we're trying to prove again simplifies to (tan2xsecx)cosx=sin2x
Now let's work on the LHS and see if we can show that it is equal to sin2x. Firstly, let's unpack the definitions of tan and sec.
(tan2x/secx)cosx=(sin2x/cos2x/1/cosx)cosx=sin2x
which is exactly the modified RHS of the equation. Thus the identity is proven.

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