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प्रश्न
`(sin theta)/((sec theta + tan theta -1)) + cos theta/((cosec theta + cot theta -1))=1`
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उत्तर
LHS = `(sin theta)/((sec theta + tan theta -1)) + cos theta/((cosec theta + cot theta -1))`
=`(sin theta cos theta)/(1+ sin theta - cos theta)+(cos theta sin theta)/(1+ cos theta - sin theta)`
=`sin theta cos theta [1/(1+ (sin theta - cos theta))+ 1/(1- (sin theta - cos theta))]`
=`sin theta cos theta [(1-(sin theta - cos theta)+1+(sin theta - cos theta))/({1+ (sin theta - cos theta )}{1- (sin theta-cos theta)})]`
=`sin theta cos theta[(1-sin theta + cos theta +1+sin theta - cos theta)/(1-(sin theta - cos theta)^2)]`
=`(2 sin theta cos theta)/(1-(sin ^2 theta + cos^2 theta -2 sin theta cos theta))`
=`(2 sin theta cos theta )/(2 sin theta cos theta)`
=1
= RHS
Hence, LHS = RHS
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संबंधित प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`
Prove the following trigonometric identities
`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
Prove the following identities:
`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`
If sec A + tan A = p, show that:
`sin A = (p^2 - 1)/(p^2 + 1)`
Prove that:
(cosec A – sin A) (sec A – cos A) sec2 A = tan A
` tan^2 theta - 1/( cos^2 theta )=-1`
`costheta/((1-tan theta))+sin^2theta/((cos theta-sintheta))=(cos theta+ sin theta)`
If x=a `cos^3 theta and y = b sin ^3 theta ," prove that " (x/a)^(2/3) + ( y/b)^(2/3) = 1.`
Write the value of ` cosec^2 (90°- theta ) - tan^2 theta`
What is the value of \[6 \tan^2 \theta - \frac{6}{\cos^2 \theta}\]
Prove the following identity :
`cos^4A - sin^4A = 2cos^2A - 1`
Prove the following identity :
`(cosA + sinA)^2 + (cosA - sinA)^2 = 2`
Without using trigonometric table , evaluate :
`(sin49^circ/sin41^circ)^2 + (cos41^circ/sin49^circ)^2`
Prove that `sin(90^circ - A).cos(90^circ - A) = tanA/(1 + tan^2A)`
Prove that `(tan θ)/(cot(90° - θ)) + (sec (90° - θ) sin (90° - θ))/(cosθ. cosec θ) = 2`.
If A = 60°, B = 30° verify that tan( A - B) = `(tan A - tan B)/(1 + tan A. tan B)`.
Prove that `(sin θ. cos (90° - θ) cos θ)/sin( 90° - θ) + (cos θ sin (90° - θ) sin θ)/(cos(90° - θ)) = 1`.
If `sec θ + tan θ = sqrt(3)`, complete the activity to find the value of sec θ – tan θ.
Activity:
`square = 1 + tan^2θ` ...[Fundamental trigonometric identity]
`square - tan^2θ = 1`
`(sec θ + tan θ) . (sec θ - tan θ) = square`
`sqrt(3) . (sec θ - tan θ) = 1`
`(sec θ - tan θ) = square`
