If P-sin 300-tan 330- 420 -tan 135 -sin 210 - 315 - and Q- 480-cosec 570-tan 330-sin 600 -cos 660 - 405- then the value of P and Q are respectively

P=sin 300^∘·tan 330^∘ nbsp;· 420 nbsp;^∘tan 135 nbsp;^∘·sin 210 nbsp;^∘· 315 nbsp;^∘ nbsp; ⇒ nbsp; P=sin (360^∘ 60^∘) ·tan (360^∘ 30^∘) · (360^∘+6 ...

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Byju's Answer

Standard XIII

Mathematics

Trigonometric Ratios of Common Angles

If P=sin 300∘...

Question

If $P = \frac{sin 300^{\circ} \cdot tan 330^{\circ} \cdot sec 420^{\circ}}{tan 135^{\circ} \cdot sin 210^{\circ} \cdot sec 315^{\circ}}$ and $Q = \frac{sec 480^{\circ} \cdot cosec 570^{\circ} \cdot tan 330^{\circ}}{sin 600^{\circ} \cdot cos 660^{\circ} \cdot cot 405^{\circ}},$ then the value of $P$ and $Q$ are respectively

\sqrt{2}, - \frac{16}{3}

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\sqrt{2}, \frac{16}{3}

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- 2, \frac{3}{16}

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\sqrt{2}, \frac{3}{16}

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Solution

The correct option is B $\sqrt{2}, \frac{16}{3}$
$P = \frac{sin 300^{\circ} \cdot tan 330^{\circ} \cdot sec 420^{\circ}}{tan 135^{\circ} \cdot sin 210^{\circ} \cdot sec 315^{\circ}}$
$\Rightarrow P = \frac{sin (360^{\circ} - 60^{\circ}) \cdot tan (360^{\circ} - 30^{\circ}) \cdot sec (360^{\circ} + 60^{\circ})}{tan (90^{\circ} + 45^{\circ}) \cdot sin (180^{\circ} + 30^{\circ}) \cdot sec (360^{\circ} - 45^{\circ})}$
$\Rightarrow P = \frac{- sin 60^{\circ} \cdot (- tan 30^{\circ}) \cdot (sec 60^{\circ})}{- cot 45^{\circ} \cdot (- sin 30^{\circ}) \cdot (sec 45^{\circ})} \Rightarrow P = \frac{(\frac{- \sqrt{3}}{2}) \cdot (\frac{- 1}{\sqrt{3}}) \cdot (2)}{(- 1) \cdot (\frac{- 1}{2}) \cdot (\sqrt{2})} \Rightarrow P = \sqrt{2}$

$Q = \frac{sec 480^{\circ} \cdot cosec 570^{\circ} \cdot tan 330^{\circ}}{sin 600^{\circ} \cdot cos 660^{\circ} \cdot cot 405^{\circ}}$
$\Rightarrow Q = \frac{sec (540^{\circ} - 60^{\circ}) \cdot cosec (540^{\circ} + 30^{\circ}) \cdot (tan 360^{\circ} - 30^{\circ})}{sin (540^{\circ} + 60^{\circ}) \cdot cos (720^{\circ} - 60^{\circ}) \cdot cot (360^{\circ} + 45^{\circ})}$
$\Rightarrow Q = \frac{(- sec 60^{\circ}) \cdot (- cosec 30^{\circ}) \cdot (- tan 30^{\circ})}{(- sin 60^{\circ}) \cdot (cos 60^{\circ}) \cdot (cot 45^{\circ})}$
$\Rightarrow Q = \frac{(- 2) (- 2) (\frac{- 1}{\sqrt{3}})}{(\frac{- \sqrt{3}}{2}) \cdot (\frac{1}{2}) \cdot (1)} \Rightarrow Q = \frac{16}{3}$

Hence, $P = \sqrt{2}, Q = \frac{16}{3}$

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Similar questions

Q. If

P = \frac{sin 300^{\circ} \cdot tan 330^{\circ} \cdot sec 420^{\circ}}{tan 135^{\circ} \cdot sin 210^{\circ} \cdot sec 315^{\circ}}

and

Q = \frac{sec 480^{\circ} \cdot cosec 570^{\circ} \cdot tan 330^{\circ}}{sin 600^{\circ} \cdot cos 660^{\circ} \cdot cot 405^{\circ}},

then the value of

P

and

Q

are respectively

\frac{s i n (- 660^{\circ}) t a n (1050^{\circ}) s e c (- 420^{\circ})}{c o s (225^{\circ}) c o s e c (315^{\circ}) c o s (510^{\circ})} =

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

Find the value of $s i n 210^{\circ} + c o s 120^{\circ} + t a n 225^{\circ} + c o t 315^{\circ} + s e c 300^{\circ} + c o s e c 150^{\circ}$ .

___

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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If P=sin300∘⋅tan330∘⋅sec420∘tan135∘⋅sin210∘⋅sec315∘ and Q=sec480∘⋅cosec 570∘⋅tan330∘sin600∘⋅cos660∘⋅cot405∘, then the value of P and Q are respectively