Product of Trigonometric Ratios in Terms of Their Sum
If cos -1√p+c...
Question
If cos−1√p+cos−1√1−p+cos−1√1−q=3π4, then the value of q is
A
1
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B
1√2
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C
13
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D
12
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Solution
The correct option is D12 Let α=cos−1√p:β=cos−1√1−p and γ=cos−1√1−q or cosα=√p:cosβ=√1−p and cosγ=√1−q. Therefore sinα=√1−p,sinβ=√p and sinγ=√q. The given equation may be written as α+β+γ=3π4orα+β=3π4−γ or cos(α+β)=cos(3π4−γ) ⇒cosαcosβ−sinαsinβ = cos{π−(π4+γ)}=−cos(π4+γ) ⇒√p√1−p−√1−p√p=−(1√2√1−q−1√2.√q) ⇒0=√1−q−√q⇒1−q=q⇒q=12.