R.S. Aggarwal solutions for Mathematics [English] Class 10 chapter 12 - Trigonometric Ratios of Some Complemantary Angles [Latest edition]

Without using trigonometric tables, evaluate :
`sin 16^circ/cos 74^circ`

Without using trigonometric tables, evaluate :

`sec 11^circ/("cosec"  79^circ)`

Without using trigonometric tables, evaluate :

`tan 27^circ/cot 63^circ`

Without using trigonometric tables, evaluate :

`cos 35^circ/sin 55^circ`

Without using trigonometric tables, evaluate :

`("cosec"  42^circ)/sec 48^circ`

Without using trigonometric tables, evaluate :

`cot 38^circ/tan 52^circ`

Without using trigonometric tables, prove that:

cos 81° − sin 9° = 0

Without using trigonometric tables, prove that:

tan 71° − cot 19° = 0

Without using trigonometric tables, prove that:

cosec 80° − sec 10° = 0

Without using trigonometric tables, prove that:

cosec²72° − tan²18° = 1

Without using trigonometric tables, prove that:

cos²75° + cos²15° = 1

Without using trigonometric tables, prove that:

tan²66° − cot²24° = 0

Without using trigonometric tables, prove that:

sin²48° + sin²42° = 1

Without using trigonometric tables, prove that:

cos²57° − sin²33° = 0

Without using trigonometric tables, prove that:

(sin 65° + cos 25°)(sin 65° − cos 25°) = 0

Without using trigonometric tables, prove that:

sin53° cos37° + cos53° sin37° = 1

Without using trigonometric tables, prove that:

cos54° cos36° − sin54° sin36° = 0

Without using trigonometric tables, prove that:

sec70° sin20° + cos20° cosec70° = 2

Without using trigonometric tables, prove that:

sin35° sin55° − cos35° cos55° = 0

Without using trigonometric tables, prove that:

(sin72° + cos18°)(sin72° − cos18°) = 0

Without using trigonometric tables, prove that:

tan48° tan23° tan42° tan67° = 1

Prove that:

`(sin 70^circ)/(cos 20^circ) + ("cosec" 20^circ)/(sec 70^circ) - 2  cos 70^circ "cosec"  20^circ = 0`

Prove that:

`cos 80^circ/(sin 10^circ) + cos 59^circ "cosec"  31^circ = 2`

Prove that:

`(2  "sin"  68^circ)/(cos 10^circ )- (2  cot 15^circ)/(5 tan 75^circ) = ((3  tan 45^circ t  an 20^circ  tan 40^circ tan 50^circ tan 70^circ)) /5= 1` 

Prove that:

`sin 18^circ/(cos 72^circ )+ sqrt(3)(tan 10^circ tan 30^circ tan 40^circ  tan50^circ tan 80^circ) `

Prove that:

sin θ cos (90° - θ ) + sin (90° - θ) cos θ = 1

Prove that:

\[\frac{\sin\theta}{\cos(90° - \theta)} + \frac{\cos\theta}{\sin(90° - \theta)} = 2\]

Prove that:

\[\frac{\sin\theta  \cos(90^\circ - \theta)\cos\theta}{\sin(90^\circ- \theta)} + \frac{\cos\theta  \sin(90^\circ - \theta)\sin\theta}{\cos(90^\circ - \theta)}\]

Prove that:

\[\frac{sin\theta  \cos(90°  - \theta)cos\theta}{\sin(90° - \theta)} + \frac{cos\theta  \sin(90° - \theta)sin\theta}{\cos(90° - \theta)}\]

Prove that:

\[\frac{\cos(90^\circ - \theta)}{1 + \sin(90^\circ - \theta)} + \frac{1 + \sin(90^\circ- \theta)}{\cos(90^\circ - \theta)} = 2 cosec\theta\]

Prove that:

\[\frac{sin\theta  \cos(90° - \theta)cos\theta}{\sin(90° - \theta)} + \frac{cos\theta  \sin(90° - \theta)sin\theta}{\cos(90° - \theta)}\]

Prove that:

\[cot\theta \tan\left( 90° - \theta \right) - \sec\left( 90° - \theta \right)cosec\theta + \sqrt{3}\tan12° \tan60° \tan78° = 2\]

Prove that :

tan5° tan25° tan30° tan65° tan85° = \[\frac{1}{\sqrt{3}}\]

Prove that:

cot12° cot38° cot52° cot60° cot78° = \[\frac{1}{\sqrt{3}}\]

Prove that:

cos15° cos35° cosec55° cos60° cosec75° = \[\frac{1}{2}\]

Prove that:

cos1° cos2° cos3° ... cos180° = 0

Prove that:

\[\left( \frac{\sin49^\circ}{\cos41^\circ} \right)^2 + \left( \frac{\cos41^\circ}{\sin49^\circ} \right)^2 = 2\]

Prove that

sin (70° + θ) − cos (20° − θ) = 0

Prove that

tan (55° − θ) − cot (35° + θ) = 0

Prove that

cosec (67° + θ) − sec (23° − θ) = 0

Prove that

 cosec (65 °+ θ)  sec  (25° −  θ) − tan (55° − θ) + cot (35° + θ) = 0

Prove that

sin (50° + θ ) − cos (40° − θ) + tan 1° tan 10° tan 80° tan 89° = 1.

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

sin67° + cos75° 

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

cot65° + tan49°

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

sec78° + cosec56°

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

cosec54° + sin72°

If A, B  and C are the angles of a  ΔABC, prove that tan `((C + "A")/2) = cot  B/2`

If cos 20 = sin 4 θ ,where 2 θ and 4 θ are acute angles, then find the value of θ

If sec2A = cosec(A - 42°), where 2A is an acute angle, then find the value of A.  

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

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

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