To prove: The identity tan θ sin θ + cos θ = sec θ .

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter T, Problem 7DDT

(a)

To determine

To prove: The identity tanθsinθ+cosθ=secθ.

Expert Solution

Explanation of Solution

Proof:

Consider the L. H. S. of the identity is tanθsinθ+cosθ. The L. H. S. can be written as,

L.H.S.=sinθcosθ×sinθ+cosθ=sin2θ+cos2θcosθ=1cosθ(sin2θ+cos2θ=1)=secθ

=R.H.S.

Thus, the identity tanθsinθ+cosθ=secθ is proved.

(b)

To determine

Expert Solution

Explanation of Solution

Proof:

Consider the L. H. S. of the identity is 2tanx1+tan2x. The L. H. S. can be written as,

L.H.S.=2sinxcosx1+sin2xcos2x=2sinxcosxcos2x+sin2xcos2x=2sinxcosx1cos2x(sin2x+cos2x=1)=2sinxcosx×cos2x

On further simplification the above expression becomes,

L.H.S.=2sinxcosx=sin2x=R.H.S.

Thus, the identity 2tanx1+tan2x=sin2x is proved.

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