Advertisements
Advertisements
Question
Prove the following trigonometric identities.
`tan theta + 1/tan theta = sec theta cosec theta`
Advertisements
Solution
We know that `sec^2 theta - tan^2 theta = 1`
So,
`tan theta + 1/tan theta = (tan^2 theta + 1)/tan theta`
`= sec^2 theta/tan theta`
`= sec theta sec theta/tan theta`
`= sec theta = (1/cos theta)/(sin theta/cos theta)`
`= sec theta cosec theta`
APPEARS IN
RELATED QUESTIONS
Prove the following identities:
`(i) cos4^4 A – cos^2 A = sin^4 A – sin^2 A`
`(ii) cot^4 A – 1 = cosec^4 A – 2cosec^2 A`
`(iii) sin^6 A + cos^6 A = 1 – 3sin^2 A cos^2 A.`
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
`(sintheta - 2sin^3theta)/(2costheta - costheta) =tan theta`
Prove the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities.
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 − y2 = a2 − b2
Prove the following identities:
`(1 + cosA)/(1 - cosA) = tan^2A/(secA - 1)^2`
Prove the following identities:
`cot^2A/(cosecA + 1)^2 = (1 - sinA)/(1 + sinA)`
If sin A + cos A = p and sec A + cosec A = q, then prove that : q(p2 – 1) = 2p.
`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`
cosec4θ − cosec2θ = cot4θ + cot2θ
`(1+ tan theta + cot theta )(sintheta - cos theta) = ((sec theta)/ (cosec^2 theta)-( cosec theta)/(sec^2 theta))`
Show that none of the following is an identity:
(i) `cos^2theta + cos theta =1`
Write True' or False' and justify your answer the following:
\[ \cos \theta = \frac{a^2 + b^2}{2ab}\]where a and b are two distinct numbers such that ab > 0.
(sec A + tan A) (1 − sin A) = ______.
Prove the following identity :
`sin^4A + cos^4A = 1 - 2sin^2Acos^2A`
If secθ + tanθ = m , secθ - tanθ = n , prove that mn = 1
Without using trigonometric table , evaluate :
`cos90^circ + sin30^circ tan45^circ cos^2 45^circ`
Evaluate:
`(tan 65^circ)/(cot 25^circ)`
Prove that: `1/(sec θ - tan θ) = sec θ + tan θ`.
Prove that `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")` = – 1
