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Precalculus Mathematics for Calculus
Found in: Page 543
Precalculus Mathematics for Calculus

Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

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Short Answer

Proving Identities

Verify the identity cos2 t+tan2 t1sin2 t=tan2 t

The expression cos2 t+tan2 t1sin2 t=tan2 t is an identity.

See the step by step solution

Step by Step Solution

Step 1. Given information.

An expression cos2 t+tan2 t1sin2 t=tan2 t

Step 2. Concept used.

To demonstrate that the preceding statement is an identity, divide it into two portions, LHS and RHS. Separate the two and simplify them independently when you've completed separating. If the results are equal, the given statement is an identity.

Step 3. Calculation.

Now, simplify LHS of cos2 t+tan2 t1sin2 t=tan2 t:

Substituting 1 by cos2 t+sin2 t,

cos2 t+tan2 t1sin2 t=cos2 t+tan2 tsin2 t+cos2 tsin2 t=tan2 tsin2 tsin2 ttan x=sin xcos x=sin2 tcos2 tsin2 tsin2 tsin2 t

cos2 t+tan2 t1sin2 t=1cos2 t11cos x=sec x=sec2 t1(sec2 x+tan2 x=1)=tan2 t

RHS of the equation is tan2 t

Both RHS and LHS are equal. Hence it is an identity

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