Simplify the expression sin^2x+tan^2x-sec^2x+cos^2x.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to simplify:(sin x)^2 + (tan x)^2 - (sec x)^2 + (cos x)^2

(sin x)^2 + (tan x)^2 - (sec x)^2 + (cos x)^2

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

(sin x)^2 -1 = - (cos x)^2

=> (sin x)^2 - (cos x)^2/(cos x)^2 + (cos x)^2

(sin x)^2 + (cos x)^2 = 1

=> 1 - 1

=> 0

The expression can be simplified to 0.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We notice that using Pythagorean identity, we can replace the sum:

(sin x)^2 + (cos x)^2 = 1

The expression will become:

1 + (tan x)^2 - (sec x)^2

But 1 + (tan x)^2 = 1/(cos x)^2

We also know that 1/(cos x)^2 = (sec x)^2

We'll re-write the expression:

1 + (tan x)^2 - (sec x)^2 = (sec x)^2 - (sec x)^2 = 0

We notice that simplifying the expression, we'll get the result: (sin x)^2 + (cos x)^2 + (tan x)^2 - (sec x)^2 = 0.

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