If $θ$ lies in the first quadrant. Which one of the following expressions is independent of $θ$ ?

sec (90^{\circ} - θ) - cot θ . cos (90^{\circ} - θ) . tan (90^{\circ} - θ)

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\frac{cot (90^{\circ} - θ)}{{cosec}^{2} θ} . \frac{sec θ . {cot}^{3} θ}{{sin}^{2} (90^{\circ} - θ)}

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cos (90^{\circ} - θ) \times cosec (90^{\circ} - θ)

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None

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Solution

The correct option is D None
A) $sec (90^{\circ} - θ) - cot θ . cos (90^{\circ} - θ) . tan (90^{\circ} - θ)$
= $c o s e c θ - cot θ sin θ cot θ$
= $\frac{1 - {cos}^{2} θ}{sin θ} = \frac{{sin}^{2} θ}{sin θ} = sin θ$

B) $\frac{cot (90^{\circ} - θ)}{c o s e c^{2} θ} . \frac{sec θ . {cot}^{3} θ}{{sin}^{2} (90^{\circ} - θ)}$

= $\frac{tan θ}{c o s e c^{2} θ} . \frac{sec θ . {cot}^{3} θ}{{cos}^{2} θ}$

= $\frac{{sin}^{3} θ}{cos θ} \frac{1}{{sin}^{3} θ}$
= $sec θ$

C) $cos (90^{\circ} - θ) \times c o s e c (90^{\circ} - θ)$
= $sin θ sec θ$
= $tan θ$

Suggest Corrections

Similar questions

Prove that:

(i) $s i n θ c o s (90^{\circ} - θ) + s i n (90^{\circ} - θ) c o s θ = 1$

(ii) $\frac{s i n θ}{c o s (90^{\circ} - θ)} + \frac{c o s θ}{s i n (90^{\circ} - θ)} = 2$

(iii) $\frac{s i n θ c o s (90^{\circ} - θ) c o s θ}{s i n (90^{\circ} - θ)} + \frac{c o s θ s i n (90^{\circ} - θ) s i n θ}{c o s (90^{\circ} - θ)} = 1$

(iv) $\frac{c o s (90^{\circ} - θ) s e c (90^{\circ} - θ) t a n θ}{c o s e c (90^{\circ} - θ) s i n (90^{\circ} - θ) c o t (90^{\circ} - θ)} + \frac{t a n (90^{\circ} - θ)}{c o t θ} = 2$

(v) $\frac{c o s (90^{\circ} - θ)}{1 + s i n (90^{\circ} - θ)} + \frac{1 + s i n (90^{\circ} - θ)}{c o s (90^{\circ} - θ)} = 2 c o s e c θ$

(vi) $\frac{s e c (90^{\circ} - θ) c o s e c θ - t a n (90^{\circ} - θ) c o t θ + c o s^{2} 25^{\circ} + c o s^{2} 65^{\circ}}{3 t a n 27^{\circ} t a n 63^{\circ}} = \frac{2}{3}$

(vii) $c o t θ t a n (90^{\circ} - θ) - s e c (90^{\circ} - θ) c o s e c θ + \sqrt{3} t a n 12^{\circ} t a n 60^{\circ} t a n 78^{\circ} = 2$

Q. Prove the following :

\frac{cos (90^{\circ} - θ) sec (90^{\circ} - θ) tan θ}{cosec (90^{\circ} - θ) sin (90^{\circ} - θ) cot (90^{\circ} - θ)} + \frac{tan (90^{\circ} - θ)}{cot θ} = 2

Which of the following options is equal to the given expresssion?

$\frac{c o t (90^{\circ} - θ)}{c o s e c^{2} θ}$ $\times$ $\frac{s e c θ . c o t^{3} θ}{s i n^{2} (90^{\circ} - θ)}$

Q. If

θ

lies in the first quadrant and

cos θ = \frac{8}{17}

, then the value of

cos (30^{o} + θ) + cos (45^{o} - θ) + cos (120^{o} - θ)

If $θ$ lies in the first quadrant and cos $θ = \frac{8}{17}$ , then the value of $c o s (30^{\circ} + θ) + c o s (45^{\circ} - θ) + c o s (120^{\circ} - θ)$ is

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If θ lies in the first quadrant. Which one of the following expressions is independent of θ?

If $θ$ lies in the first quadrant. Which one of the following expressions is independent of $θ$ ?