"How to prove the following: tan2x+1+tan⁡xsec⁡x=1+sin⁡xcos2x" solved and explained

Cem Hayes

Answered question

2021-03-02

How to prove the following:

\tan^{2} x + 1 + \tan x \sec x = 1 + \frac{\sin x}{\cos^{2} x}

hosentak

Skilled2021-03-03Added 100 answers

\tan x \cdot \sec x = \frac{\sin x}{\cos^{2} x} or \sin x \cdot \sec^{2} x . \tan^{2} x + 1 = \sec^{2} x .

So we have

\sec^{2} x + \sec^{2} x \sin x = \sec^{2} x (1 + \sin x) = \frac{1 + \sin x}{\cos^{2} x} .

(To prove

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

, start with

\sin^{2} x + \cos^{2} x = 1

, then divide through by

\cos^{2} x : \tan^{2} x + 1 = \sec^{2} x

because secx is

\frac{1}{\cos x}

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