Advertisements
Advertisements
Question
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`
Solution
`((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A`
`(1+tan^2A)/(1+cot^2A)=(1+sin^2A/cos^2A)/(1+cos^2A/sin^2A)`
= `((cos^2A + sin^2A)/cos^2A)/((sin^2A + cos^2A)/sin^2A)`
= `(1/cos^2A)/(1/sin^2A)`
= `sin^2A/cos^2A`
= tan2A
`((1-tanA)/(1-cotA))^2=(1+tan^2A-2tanA)/(1+cot^2A-2cotA)`
= `(sec^2A-2tanA)/(cosec^2A-2cotA)`
= `(1/cos^2A-(2sinA)/cosA)/(1/sin^2A-(2cosA)/sinA)`
= `((1 - 2sinAcosA)/cos^2A)/((1 - 2sinAcosA)/sin^2A)`
= `sin^2A/cos^2A`
= tan2A
APPEARS IN
RELATED QUESTIONS
Without using trigonometric tables, evaluate
`sin^2 34^@ + sin^2 56^@ + 2tan 18^@ tan 72^@ - cot^2 30^@`
Without using tables evaluate: 3cos 80°. cosec 10° + 2sin 59° sec 31°
Without using trigonometric tables, evaluate :
`sec 11^circ/("cosec" 79^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:
cosec272° − tan218° = 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:
(sin 65° + cos 25°)(sin 65° − cos 25°) = 0
Without using trigonometric tables, prove that:
sin35° sin55° − cos35° cos55° = 0
Without using trigonometric tables, prove that:
(sin72° + cos18°)(sin72° − cos18°) = 0
Prove that:
`cos 80^circ/(sin 10^circ) + cos 59^circ "cosec" 31^circ = 2`
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:
\[\frac{\cos(90^\circ - \theta)}{1 + \sin(90^\circ - \theta)} + \frac{1 + \sin(90^\circ- \theta)}{\cos(90^\circ - \theta)} = 2 cosec\theta\]
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.
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.
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]
Solve : Sin2θ - 3sin θ + 2 = 0 .
From trigonometric table, write the values of sin 37°19'.
From the trigonometric table, write the values of cos 23°17'.
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`.
Using trigonometric table evaluate the following:
sin 64°42' + cos 42°20'
Using trigonometric table evaluate the following:
cos 64°42' - sin 42°20'
Using trigonometric table evaluate the following:
tan 78°55' - tan 55°18'
`(sin 40° + cos 50°)/(tan 38°20')`
`(sin 20°50' + tan 67°40')/(cos 32°20' - sin 15°10')`
The maximum value of `1/(cosec alpha)` is ______.
Prove that:
`(cos^2 "A")/(cos "A" - sin "A") + (sin "A")/(1 - cot "A")` = sin A + cos A
If sin θ = 1, then the value of `1/2 sin(theta/2)`is ______.
