Rs Aggarwal 2015 Solutions for Class 10 Math Chapter 1 Real Numbers are provided here with simple step-by-step explanations. These solutions for Real Numbers are extremely popular among Class 10 students for Math Real Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2015 Book of Class 10 Math Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnationâ€™s Rs Aggarwal 2015 Solutions. All Rs Aggarwal 2015 Solutions for class Class 10 Math are prepared by experts and are 100% accurate.

#### Question 1:

Without using trigonometric tables, evaluate:
(i)
(ii)
(iii)
(iv)
(v)
(vi)

#### Question 2:

Prove that:

(i)
(ii) $\frac{\mathrm{sin\theta }}{\mathrm{cos}\left(90°-\mathrm{\theta }\right)}+\frac{\mathrm{cos\theta }}{\mathrm{sin}\left(90°-\mathrm{\theta }\right)}=2$
(iii)
(iv) $\frac{\mathrm{cos}\left(90°-\mathrm{\theta }\right)\mathrm{sec}\left(90°-\mathrm{\theta }\right)\mathrm{tan\theta }}{\mathrm{cosec}\left(90°-\mathrm{\theta }\right)\mathrm{sin}\left(90°-\mathrm{\theta }\right)\mathrm{cot}\left(90°-\mathrm{\theta }\right)}+\frac{\mathrm{tan}\left(90°-\mathrm{\theta }\right)}{\mathrm{cot\theta }}=2$
(v) $\frac{\mathrm{cos}\left(90°-\mathrm{\theta }\right)}{1+\mathrm{sin}\left(90°-\mathrm{\theta }\right)}+\frac{1+\mathrm{sin}\left(90°-\mathrm{\theta }\right)}{\mathrm{cos}\left(90°-\mathrm{\theta }\right)}=2\mathrm{cosec\theta }$
(vi)
(vii)

#### Question 3:

Without using trigonometric tables, evaluate:
(i)
(ii)
(iii)
(iv)

#### Question 4:

Without using trigonometric tables, prove that:
(i) cos 81° − sin 9° = 0
(ii) tan 71° − cot 19° = 0
(iii) cosec 80° − sec 10° = 0
(iv) cosec272° − tan218° = 1
(v) cos275° + cos215° = 1
(vi) tan266° − cot224° = 0
(vii) sin248° + sin242° = 1
(viii) cos257° − sin233° = 0
(ix) (sin 65° + cos 25°)(sin 65° − cos 25°) = 0

#### Question 5:

Without using trigonometric tables, prove that:

(i) sin53° cos37° + cos53° sin37° = 1
(ii) cos54° cos36° − sin54° sin36° = 0
(iii) sec70° sin20° + cos20° cosec70° = 2
(iv) sin35° sin55° − cos35° cos55° = 0
(v) (sin72° + cos18°)(sin72° − cos18°) = 0
(vi) tan48° tan23° tan42° tan67° = 1

#### Question 6:

Prove that:

(i) tan5° tan25° tan30° tan65° tan85° = $\frac{1}{\sqrt{3}}$
(ii) cot12° cot38° cot52° cot60° cot78° = $\frac{1}{\sqrt{3}}$
(iii) cos15° cos35° cosec55° cos60° cosec75° = $\frac{1}{2}$
(iv) cos1° cos2° cos3° ... cos180° = 0
(v) ${\left(\frac{\mathrm{sin}49°}{\mathrm{cos}41°}\right)}^{2}+{\left(\frac{\mathrm{cos}41°}{\mathrm{sin}49°}\right)}^{2}=2$

Disclaimer: The RHS of (v) given in textbook is incorrect. There should be 2 instead 1. The same has been corrected in the solution here.

#### Question 7:

Prove that
(i) sin (70° + θ) − cos (20° − θ) = 0
(ii) tan (55° − θ) − cot (35° + θ) = 0
(iii) cosec (67° + θ) − sec (23° − θ) = 0
(iv) cosec (65 °+ θ)  sec  (25° −  θ) − tan (55° − θ) + cot (35° + θ) = 0
(v) sin (50° + θ ) − cos (40° − θ) + tan 1° tan 10° tan 80° tan 89° = 1.

#### Question 8:

Express each of the following in terms of trigonometric ratios of angles lying between 0° and 45°.

(i) sin67° + cos75°
(ii) cot65° + tan49°
(iii) sec78° + cosec56°
(iv) cosec54° + sin72°

#### Question 9:

If sin 3 A = cos (A − 26°), where 3 A is an acute angle, find the value of A.

#### Question 10:

If tan 2 A = cot (A − 12°), where 2 A is an acute angle, find the value of A.

#### Question 11:

If sec 4 A = cosec (A − 15°), where 4 A is an acute angle, find the value of A.

#### Question 12:

If A, B, C are the angles of a ΔABC, prove that tan ($\frac{B+C}{2}$) = cot $\frac{A}{2}$.

Prove that:

Hence Proved

#### Question 19:

tan 7° tan 23° tan 60° tan 67° tan 83° + $\frac{\mathrm{cot}54°}{\mathrm{tan}36°}$ + sin 20° sec 70° − 2

#### Question 20:

(sin225° + sin265°) + $\sqrt{3}$ (tan 5° tan 15° tan 30° tan 75° tan 85°)

#### Question 24:

$\frac{{\mathrm{sec}}^{2}\mathrm{\theta }-{\mathrm{cot}}^{2}\left(90°-\mathrm{\theta }\right)}{{\mathrm{cosec}}^{2}67°-{\mathrm{tan}}^{2}23°}+\left({\mathrm{sin}}^{2}40°+{\mathrm{sin}}^{2}50°\right)$

#### Question 1:

$\frac{\mathrm{tan}30°}{\mathrm{cot}60°}=?$
(a) $\frac{1}{\sqrt{2}}$
(b) $\frac{1}{\sqrt{3}}$
(c) $\sqrt{3}$
(d) 1

(d) 1

#### Question 2:

$\frac{\mathrm{tan}35°}{\mathrm{cot}55°}+\frac{\mathrm{cot}78°}{\mathrm{tan}12°}=?$
(a) 0
(b) 1
(c) 2
(d) None of these

#### Question 3:

${\left(\frac{\mathrm{tan}25°}{\mathrm{cosec}65°}\right)}^{2}+{\left(\frac{\mathrm{cot}25°}{\mathrm{sec}25°}\right)}^{2}=?$
(a) 1
(b) 2
(c) $\frac{3}{4}$
(d) 8

#### Question 4:

cos 1° cos 2° cos 3° ... cos 180° = ?
(a) −1
(b) 1
(c) 0
(d) $\frac{1}{2}$

#### Question 5:

tan 10° tan 15° tan 75° tan 80° = ?
(a) $\sqrt{3}$
(b) $\frac{1}{\sqrt{3}}$
(c) −1
(d) 1

#### Question 6:

tan 1° tan 2° tan 3° ... tan 89° = ?
(a) 1
(b) 0
(c) Cannot be determined
(d) None of these

#### Question 7:

tan 5° tan 25° tan 30° tan 65° tan 85° = ?
(a) $\sqrt{3}$
(b) $\frac{1}{\sqrt{3}}$
(c) 1
(d) none of these

#### Question 8:

cot 15° cot 16° cot 17° ... cot 73° cot 74° cot 75° = ?
(a) $\frac{1}{2}$
(b) 0
(c) 1
(d) −1

#### Question 9:

$\frac{2{\mathrm{sin}}^{2}63°+1+2{\mathrm{sin}}^{2}27°}{3{\mathrm{cos}}^{2}17°-2+3{\mathrm{cos}}^{2}73°}=?$
(a) $\frac{3}{2}$
(b) $\frac{2}{3}$
(c) 2
(d) 3

#### Question 10:

(sin 40° − cos 50°) = ?
(a) sin 10°
(b) cos 10°
(c) 1
(d) 0

#### Question 11:

(cos237° − sin253°) = ?
(a) $\frac{1}{3}$
(b) $\frac{2}{\sqrt{3}}$
(c) 1
(d) 0

#### Question 12:

(sin 43°cos 47° + cos 43°sin 47°) = ?
(a) sin 4°
(b) cos 4°
(c) 1
(d) 0

#### Question 13:

$\left(\frac{\mathrm{sin}49°}{\mathrm{cos}41°}-\frac{\mathrm{cos}17°}{\mathrm{sin}73°}\right)=?$
(a) 1
(b) 0
(c) −1
(d) Cannot be calculated

#### Question 14:

(sin 79° cos 11° + cos 79° sin 11°) = ?
(a) $\frac{1}{\sqrt{2}}$
(b) $\frac{1}{2}$
(c) 0
(d) 1

#### Question 15:

$\frac{\mathrm{cot}\left(90°-\mathrm{\theta }\right)·\mathrm{sin}\left(90°-\mathrm{\theta }\right)}{\mathrm{sin\theta }}+\frac{\mathrm{cot}40°}{\mathrm{tan}50°}-\left({\mathrm{cos}}^{2}20°+{\mathrm{cos}}^{2}70°\right)=?$
(a) 3
(b) 2
(c) 1
(d) 0

#### Question 16:

$\frac{2{\mathrm{tan}}^{2}30°{\mathrm{sec}}^{2}52°{\mathrm{sin}}^{2}38°}{\left({\mathrm{cosec}}^{2}70°-{\mathrm{tan}}^{2}20°\right)}=?$
(a) $\frac{3}{2}$
(b) $\frac{2}{3}$
(c) 2
(d) $\frac{1}{2}$

(b) $\frac{2}{3}$

#### Question 17:

$\left\{\frac{\left({\mathrm{sin}}^{2}22°+{\mathrm{sin}}^{2}68°\right)}{\left({\mathrm{cos}}^{2}22°+{\mathrm{cos}}^{2}68°\right)}+{\mathrm{sin}}^{2}63°+\mathrm{cos}63°\mathrm{sin}27\right\}=?$
(a) 0
(b) 1
(c) 2
(d) 3

#### Question 18:

(cosec257° − tan233°) = ?
(a) 0
(b) 1
(c) 2
(d) None of these

#### Question 19:

(cos228° − sin262°) = ?
(a) 0
(b) 1
(c) 2
(d) None of these

#### Question 20:

(sec210° − cot280°) = ?
(a) 1
(b) 0
(c) $\frac{3}{2}$
(d) None of these

#### Question 21:

cos (40° + θ) − sin (50° − θ) = ?
(a) 1
(b) 0
(c) sin 2θ
(d) None of these

#### Question 22:

sin (45° + θ) − cos (45° − θ) = ?
(a) 2 sin θ
(b) 2 cos θ
(c) 0
(d) 1

#### Question 23:

cosec (75° + θ) −sec (15° − θ) = ?
(a) 2 sec θ
(b) 2 cosec θ
(c) 0
(d) 1

#### Question 24:

If sin (θ + 34°) = cos θ and θ is acute, then θ = ?
(a) 56°
(b) 66°
(c) 28°
(d) 42°

(d) 42°

#### Question 25:

sin θ cos (90° − θ) + cos θ sin (90° − θ) = ?
(a) 0
(b) 1
(c) 2
(d) $\frac{3}{2}$

(b) 1

#### Question 26:

(a) $\sqrt{3}$
(b) $\frac{1}{3}$
(c) $\frac{1}{\sqrt{3}}$
(d) $\frac{2}{\sqrt{3}}$

(c) $\frac{1}{\sqrt{3}}$

#### Question 27:

If sin 3A = cos (A − 10°), where 3A is an acute angle, then ∠A = ?
(a) 35°
(b) 25°
(c) 20°
(d) 45°

#### Question 28:

If tan 2A = cot (A − 21°), where 2A is an acute angle, then ∠A = ?
(a) 24°
(b) 27°
(c) 35°
(d) 37°

(d) 37°

#### Question 29:

If sec 5A = cosec (A − 30°), where 5A is an acute angle, then ∠A = ?
(a) 35°
(b) 25°
(c) 20°
(d) 27°

#### Question 30:

If A and B are acute angles such that sin A = cos B, then (A + B) = ?
(a) 45°
(b) 60°
(c) 90°
(d) 180°

#### Question 31:

sec 70° sin 20° + cos 20° cosec 70° = ?
(a) 0
(b) 1
(c) −1
(d) 2

#### Question 32:

cot (90° − θ) = ?
(a) cot θ
(b) −cot θ
(c) tan θ
(d) −tan θ

$\left(c\right) tan \theta \phantom{\rule{0ex}{0ex}}\text{cot}\left({90}^{0}-\theta \right)=\mathrm{tan}\theta$

#### Question 33:

If cos 9 α = sin α and 9α < 90°, then tan 5 α = ?
(a) $\frac{1}{\sqrt{3}}$
(b)  $\sqrt{3}$
(c) 1
(d) 0

(c) 1

#### Question 34:

If cos(α + β) = 0, then sin(α − β) = ?
(a) sin α
(b) cos β
(c) sin 2α
(d) cos 2β

(d) cos 2β

#### Question 1:

(a) $3\frac{1}{2}$
(b) 4
(c) 6
(d) 5

(b) 4

#### Question 2:

The value of $\left({\mathrm{sin}}^{2}30°{\mathrm{cos}}^{2}45°+4{\mathrm{tan}}^{2}30°+\frac{1}{2}{\mathrm{sin}}^{2}90°+\frac{1}{8}{\mathrm{cot}}^{2}60°\right)=?$
(a) $\frac{3}{8}$
(b) $\frac{5}{8}$
(c) 6
(d) 2

(d) 2

#### Question 3:

If cos A + cos2 A = 1, then (sin2 A + sin4 A) = ?
(a) $\frac{1}{2}$
(b) 2
(c) 1
(d) 4

#### Question 4:

If sin $\mathrm{\theta }=\frac{\sqrt{3}}{2}$, then (cosec θ + cot θ) = ?
(a) $\left(2+\sqrt{3}\right)$
(b) $2\sqrt{3}$
(c) $\sqrt{2}$
(d) $\sqrt{3}$

(d) $\sqrt{3}$

#### Question 5:

If cot $A=\frac{4}{5}$, prove that $\frac{\left(\mathrm{sin}A+\mathrm{cos}A\right)}{\left(\mathrm{sin}A-\mathrm{cos}A\right)}=9.$

#### Question 6:

If 2x = sec A and $\frac{2}{x}$ = tan A, prove that $\left({x}^{2}-\frac{1}{{x}^{2}}\right)=\frac{1}{4}.$

#### Question 7:

If $\sqrt{3}$ tan θ = 3 sin θ, prove that (sin2 θ − cos2 θ) = $\frac{1}{3}.$

#### Question 8:

Prove that $\frac{\left({\mathrm{sin}}^{2}73°+{\mathrm{sin}}^{2}17°\right)}{\left({\mathrm{cos}}^{2}28°+{\mathrm{cos}}^{2}62°\right)}=1.$

$\frac{\left({\mathrm{sin}}^{2}73°+{\mathrm{sin}}^{2}17°\right)}{\left({\mathrm{cos}}^{2}28°+{\mathrm{cos}}^{2}62°\right)}=1.$

#### Question 9:

If 2 sin 2θ =$\sqrt{3}$, prove that θ = 30°.

#### Question 10:

Prove that  = (cosec A + cot A).

= (cosec A + cot A).

#### Question 11:

If cosec θ + cot θ = p, prove that cos θ = $\frac{\left({p}^{2}-1\right)}{\left({p}^{2}+1\right)}.$

#### Question 12:

Prove that (cosec A − cot A)2 =

(cosec A − cot A)2 =

#### Question 13:

If 5 cot θ = 3, find the value of $\left(\frac{5\mathrm{sin\theta }-3\mathrm{cos\theta }}{4\mathrm{sin\theta }+3\mathrm{cos\theta }}\right).$

#### Question 14:

Prove that (sin 32° cos 58° + cos 32° sin 58°) = 1.

(sin 32° cos 58° + cos 32° sin 58°) = 1

#### Question 15:

If x = a sin θ + b cos 0 and y = a cos 0 b sin θ, prove that ${x}^{2}+{y}^{2}={a}^{2}+{b}^{2}.$

#### Question 16:

Prove that $\frac{\left(1+\mathrm{sin\theta }\right)}{\left(1-\mathrm{sin\theta }\right)}$ = (sec θ + tan θ)2.

$\frac{\left(1+\mathrm{sin\theta }\right)}{\left(1-\mathrm{sin\theta }\right)}$= (sec θ + tan θ)2

#### Question 17:

Prove that $\frac{1}{\left(\mathrm{sec\theta }-\mathrm{tan\theta }\right)}-\frac{1}{\mathrm{cos\theta }}=\frac{1}{\mathrm{cos\theta }}-\frac{1}{\left(\mathrm{sec\theta }+\mathrm{tan\theta }\right)}$.

$\frac{1}{\left(\mathrm{sec\theta }-\mathrm{tan\theta }\right)}-\frac{1}{\mathrm{cos\theta }}=\frac{1}{\mathrm{cos\theta }}-\frac{1}{\left(\mathrm{sec\theta }+\mathrm{tan\theta }\right)}$

Prove that

Prove that