Advertisements
Advertisements
Question
`sin theta (1+ tan theta) + cos theta (1+ cot theta) = ( sectheta+ cosec theta)`
Advertisements
Solution 1
LHS = ` sin theta (1+ tan theta ) + cos theta ( 1+ cot theta )`
=` sin theta + sin theta xx (sin theta)/(cos theta) + cos theta +cos theta xx (cos theta)/( sin theta)`
= `( cos theta sin ^2 theta + sin^3 theta + cos^2 theta sin theta + cos^3 theta)/(cos theta sin theta)`
=`((sin^3 theta + cos^3 theta)+(cos theta sin ^2 theta + cos ^2 theta sin theta))/(cos theta sin theta)`
=`((sin theta + cos theta )(sin^2 theta - sin theta cos theta + cos ^2 theta )+ sin theta cos theta ( sin theta + cos theta))/(cos theta sin theta)`
=`((sin theta + cos theta )( sin^2 theta + cos^2 theta - sin theta cos theta + sin theta cos theta))/(cos theta sin theta)`
=`((sin theta + cos theta)(1))/(cos theta sin theta)`
= `(sin theta)/(cos theta sin theta) + (cos theta)/( cos theta sin theta)`
=`1/cos theta + 1/ sin theta`
=` sec theta + cosec theta`
=RHS
Solution 2
LHS = ` sin theta (1+ tan theta ) + cos theta ( 1+ cot theta )`
=` sin theta + sin theta xx (sin theta)/(cos theta) + cos theta +cos theta xx (cos theta)/( sin theta)`
= `( cos theta sin ^2 theta + sin^3 theta + cos^2 theta sin theta + cos^3 theta)/(cos theta sin theta)`
=`((sin^3 theta + cos^3 theta)+(cos theta sin ^2 theta + cos ^2 theta sin theta))/(cos theta sin theta)`
=`((sin theta + cos theta )(sin^2 theta - sin theta cos theta + cos ^2 theta )+ sin theta cos theta ( sin theta + cos theta))/(cos theta sin theta)`
=`((sin theta + cos theta )( sin^2 theta + cos^2 theta - sin theta cos theta + sin theta cos theta))/(cos theta sin theta)`
=`((sin theta + cos theta)(1))/(cos theta sin theta)`
= `(sin theta)/(cos theta sin theta) + (cos theta)/( cos theta sin theta)`
=`1/cos theta + 1/ sin theta`
=` sec theta + cosec theta`
=RHS
APPEARS IN
RELATED QUESTIONS
Prove the following identities:
`(i) 2 (sin^6 θ + cos^6 θ) –3(sin^4 θ + cos^4 θ) + 1 = 0`
`(ii) (sin^8 θ – cos^8 θ) = (sin^2 θ – cos^2 θ) (1 – 2sin^2 θ cos^2 θ)`
(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA ` using the identity cosec2 A = 1 cot2 A.
Prove the following trigonometric identities.
`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`
Prove the following identities:
sec2 A . cosec2 A = tan2 A + cot2 A + 2
Prove the following identities:
`sqrt((1 + sinA)/(1 - sinA)) = sec A + tan A`
Prove the following identities:
`sqrt((1 - cosA)/(1 + cosA)) = sinA/(1 + cosA)`
If sec θ + tan θ = x, then sec θ =
Prove the following identity :
( 1 + cotθ - cosecθ) ( 1 + tanθ + secθ)
Prove the following identity :
`cosA/(1 - tanA) + sinA/(1 - cotA) = sinA + cosA`
If m = a secA + b tanA and n = a tanA + b secA , prove that m2 - n2 = a2 - b2
Evaluate:
sin2 34° + sin2 56° + 2 tan 18° tan 72° – cot2 30°
Prove that: `(sec θ - tan θ)/(sec θ + tan θ ) = 1 - 2 sec θ.tan θ + 2 tan^2θ`
Prove that: 2(sin6θ + cos6θ) - 3 ( sin4θ + cos4θ) + 1 = 0.
Prove the following identities: cot θ - tan θ = `(2 cos^2 θ - 1)/(sin θ cos θ)`.
Prove the following identities.
tan4 θ + tan2 θ = sec4 θ – sec2 θ
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
If `sqrt(3)` sin θ – cos θ = θ, then show that tan 3θ = `(3tan theta - tan^3 theta)/(1 - 3 tan^2 theta)`
Prove that `cot^2 "A" [(sec "A" - 1)/(1 + sin "A")] + sec^2 "A" [(sin"A" - 1)/(1 + sec"A")]` = 0
Prove that `(sintheta + tantheta)/cos theta` = tan θ(1 + sec θ)
