Fill in the blanksi tan 48-tan 23- - tan 42-tan 67-ii cos 38-cos 52-sin 38-sin 52-

I) tan48^∘.tan23^∘.tan42^∘.tan67^∘ We know that cot(90^∘ θ) = tanθ ∴ tan48^∘.tan23^∘.tan42^∘.tan67^∘ = tan48^∘.tan23^∘.cot(90^∘ 42^∘).tan(90^∘ 67^∘) = tan ...

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Byju's Answer

Standard IX

Mathematics

Complementary Trigonometric Ratios

Fill in the b...

Question

Fill in the blanks
i) $t a n 48^{\circ} . t a n 23^{\circ} . t a n 42^{\circ} . t a n 67^{\circ}$ = _
ii) $c o s 38^{\circ} . c o s 52^{\circ} - s i n 38^{\circ} . s i n 52^{\circ}$ = _

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Solution

i) $t a n 48^{\circ} . t a n 23^{\circ} . t a n 42^{\circ} . t a n 67^{\circ}$

We know that $c o t (90^{\circ} - θ) = t a n θ$

$∴ t a n 48^{\circ} . t a n 23^{\circ} . t a n 42^{\circ} . t a n 67^{\circ}$
= $t a n 48^{\circ} . t a n 23^{\circ} . c o t (90^{\circ} - 42^{\circ}) . t a n (90^{\circ} - 67^{\circ})$
= $t a n 48^{\circ} . t a n 23^{\circ} . c o t 48^{\circ} . c o t 23^{\circ}$

We also know that $t a n θ \times c o t θ = 1$

$∴ t a n 48^{\circ} . t a n 23^{\circ} . c o t 48^{\circ} . c o t 23^{\circ}$
= $t a n 48^{\circ} . c o t 48^{\circ} . t a n 23^{\circ} . c o t 23^{\circ}$
= $1 \times 1$
= 1

ii) $c o s 38^{\circ} . c o s 52^{\circ} - s i n 38^{\circ} . s i n 52^{\circ}$

We know that $c o s θ = s i n (90^{\circ} - θ)$ and $s i n θ = c o s (90^{\circ} - θ)$

$∴ c o s 38^{\circ} . c o s 52^{\circ} - s i n 38^{\circ} . s i n 52^{\circ}$
= $c o s 38^{\circ} . s i n (90^{\circ} - 52^{\circ}) - s i n 38^{\circ} . c o s (90^{\circ} - 52^{\circ}$
= $c o s 38^{\circ} . s i n 38^{\circ} - s i n 38^{\circ} . c o s 38^{\circ}$
= $0$

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Similar questions

Q. Show that : (i)

tan 48^{\circ} tan 23^{\circ} tan 42^{\circ} tan 67^{\circ} = 1

(ii)

cos 38^{\circ} cos 52^{\circ} - sin 38^{\circ} sin 52^{\circ} = 0

Q. The value of

t a n 48^{\circ} . t a n 23^{\circ} . t a n 42^{\circ} . t a n 67^{\circ}

and

c o s 38^{\circ} . c o s 52^{\circ} - s i n 38^{\circ} . s i n 52^{\circ}

is equal to

Q. Simplify

cos 38^{\circ} cos 52^{\circ} - sin 38^{\circ} sin 52^{\circ}

Q. Question 2 (ii)
Show that :

c o s 38^{\circ} c o s 52^{\circ} - s i n 38^{\circ} s i n 52^{\circ} = 0

Q. show that

cos 38^{\circ} cos 52^{\circ} - sin 38^{\circ} sin 52^{\circ} = 0

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Fill in the blanks i) tan 48∘.tan 23∘.tan 42∘.tan 67∘ = ___ ii) cos 38∘.cos 52∘−sin 38∘.sin 52∘ = ___

Fill in the blanks
i) $t a n 48^{\circ} . t a n 23^{\circ} . t a n 42^{\circ} . t a n 67^{\circ}$ = _
ii) $c o s 38^{\circ} . c o s 52^{\circ} - s i n 38^{\circ} . s i n 52^{\circ}$ = _