Simplify and find the value of \tan\theta(1-\sec\frac{\theta}{2})(1-\sec\theta)(1-\sec2\theta)...(1--\sec2^{n-1}\theta) at n=1,2,3

Eileen Garza

Eileen Garza

Answered question

2022-04-26

Simplify and find the value of
tanθ(1secθ2)(1secθ)(1sec2θ)(1sec2n1θ)
at n=1,2,3

Answer & Explanation

utloverej

utloverej

Beginner2022-04-27Added 15 answers

Use the following identity:
1sec(2mθ)=(1cos(2mθ))(sec(2mθ))=2cos(2m1θ)cos(2mθ)
The given equation becomes:
tan(θ)(2)n+1cos(14θ)cos(2n1θ)
If n=1, 4cos(θ2)tan(θ)sec(θ)
If n=2, 8cos(θ4)tan(θ)sec(2θ)
If n=3, 16cos(θ4)tan(θ)sec(4θ)

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