Advertisements
Advertisements
प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
(cosec A - sin A) (sec A - cos A) = `1/(tanA+cotA)`
[Hint: Simplify LHS and RHS separately.]
उत्तर
(cosec A – sin A) (sec A – cos A) = `1/(tanA+cotA)`
L.H.S. = (cosec A – sin A) (sec A – cos A)
= `(1/sinA-sinA)(1/cosA-cosA)`
= `((1-sin^2A)/sinA)((1-cos^2A)/cosA)`
= `((cos^2A)(sin^2A))/(sinAcosA)`
= sinA cosA
R.H.S = `1/(tanA+cotA)`
= `1/(sinA/cosA+cosA/sinA)`
= `1/((sin^2A + cos^2A)/(sinAcosA))`
= `(sinAcosA)/(sin^2A+cos^2A)`
= sinA cosA
Hence, L.H.S = R.H.S
APPEARS IN
संबंधित प्रश्न
Without using trigonometric tables, evaluate
`sin^2 34^@ + sin^2 56^@ + 2tan 18^@ tan 72^@ - cot^2 30^@`
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A`
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 :
`cos 35^circ/sin 55^circ`
Without using trigonometric tables, evaluate :
`("cosec" 42^circ)/sec 48^circ`
Without using trigonometric tables, prove that:
cos 81° − sin 9° = 0
Without using trigonometric tables, prove that:
cosec 80° − sec 10° = 0
Without using trigonometric tables, prove that:
tan266° − cot224° = 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:
sin35° sin55° − cos35° cos55° = 0
Without using trigonometric tables, prove that:
tan48° tan23° tan42° tan67° = 1
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:
\[\frac{\cos(90^\circ - \theta)}{1 + \sin(90^\circ - \theta)} + \frac{1 + \sin(90^\circ- \theta)}{\cos(90^\circ - \theta)} = 2 cosec\theta\]
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:
cot12° cot38° cot52° cot60° cot78° = \[\frac{1}{\sqrt{3}}\]
If sin 3 A = cos (A − 26°), where 3 A is an acute angle, find the value of A.
Prove that `(sin "A" - cos "A" + 1)/(sin "A" + cos "A" - 1) = 1/(sec "A" - tan "A")`
A man in a boat rowing away from a lighthouse 100 m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from 60° to 30°. Find the speed of the boat in metres per minute [Use `sqrt3` = 1.732]
Prove the following:
`1/(1+sin^2theta) + 1/(1+cos^2theta) + 1/(1+sec^2theta) + 1/(1+cosec^2theta) = 2`
Given that `tan (θ_1 + θ_2) = (tan θ_1 + tan θ_2)/(1 - tan θ_1 tan θ_2)` Find (θ1 + θ2) when tan θ1 = `1/2 and tan θ_2 = 1/3`.
Without using trigonometric tables, find the value of (sin 72° + cos 18°)(sin 72° - cos 18°).
Solve : Sin2θ - 3sin θ + 2 = 0 .
From the trigonometric table, write the values of tan 45°48'.
Solve the following equation: `(cos θ)/(1 - sin θ) + cos θ/(1 + sinθ) = 4`.
Solve the following equation: `(cos^2θ - 3 cosθ + 2)/sin^2θ` = 1.
`(sin 40° + cos 50°)/(tan 38°20')`
Given that sin θ = `a/b` then cos θ is equal to ______.
