Question

# The value of tan 48∘.tan 23∘.tan 42∘.tan 67∘ and cos 38∘.cos 52∘−sin 38∘.sin 52∘ is equal to

A
1 and 0 respectively
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B
1 and 1 respectively
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C
1 and 2 respectively
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D
2 and 1 respectively
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Solution

## The correct option is A 1 and 0 respectivelyi) tan48∘.tan23∘.tan42∘.tan67∘ We know that, cot(90∘−θ)=tanθ ∴tan48∘.tan23∘.tan42∘.tan67∘ = tan48∘.tan23∘.cot(90∘−42∘).tan(90∘−67∘) = tan48∘.tan23∘.cot48∘.cot23∘ We also know that, tanθ×cotθ=1 ∴tan48∘.tan23∘.cot48∘.cot23∘ = tan48∘.cot48∘.tan23∘.cot23∘ = 1×1 = 1 ii) cos38∘.cos52∘−sin38∘.sin52∘ We know that, cosθ=sin(90∘−θ) and cosθ=sin(90∘−θ) ∴cos38∘.cos52∘−sin38∘.sin52∘ = cos38∘.sin(90∘−52∘)−sin38∘.cos(90∘−52∘ = cos38∘.sin38∘−sin38∘.cos38∘ = 0

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