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Question
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
Options
sin2β
sin4β
sin3β
cos2β
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Solution
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to sin4β.
Explanation:
Given that: tanα = `1/7`, tanβ = `1/3`
cos2α = `(1 - tan^2 alpha)/(1 + tan^2 alpha)`
= `(1 - (1/7)^2)/(1 + (1/7)^2)`
= `(1 - 1/49)/(1 + 1/49)`
= `48/50`
= `24/25`
Now tan2β = `(2tan beta)/(1 - tan^2 beta)`
= `(2 xx 1/3)/(1 - 1/9)`
= `(2/3)/(8/9)`
= `2/3 xx 9/8`
= `3/4`
∴ tan2β = `3/4`
sin4β = `(2tan 2beta)/(1 + tan^2 2beta)`
= `(2 xx 3/4)/(1 + (3/4)^2`
= `(3/2)/(1 + 9/16)`
= `3/2 xx 16/25`
= `24/25`
cos2α = sin4β = `24/25`
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