Advertisements
Advertisements
प्रश्न
`{1/((sec^2 theta- cos^2 theta))+ 1/((cosec^2 theta - sin^2 theta))} ( sin^2 theta cos^2 theta) = (1- sin^2 theta cos ^2 theta)/(2+ sin^2 theta cos^2 theta)`
उत्तर
LHS = `{1/((sec^2 theta- cos^2 theta))+ 1/((cosec^2 theta - sin^2 theta))} ( sin^2 theta cos^2 theta) `
=`{(cos^2 theta)/(1- cos^4 theta)+ (sin^2 theta)/(1- sin^4 theta)}(sin^2 theta cos ^2 theta)`
=`{cos^2 theta/((1-cos^2 theta)(1+ cos^2 theta)) + sin^2 theta/((1-sin^2 theta)(2+ sin^2 theta ))}(sin^2 theta cos^2 theta)`
=`[cot^2 theta/(1+ cos^2 theta) + tan^2 theta/(1+ sin^2 theta)]sin^2 theta cos^2 theta`
=`(cos^4 theta)/(1+ cos^2 theta)+( sin^4 theta) / (1+ sin^2 theta)`
=`((cos^2 theta)^2)/(1+ cos^2 theta)+ ((sin^2 theta)^2)/(1+ sin^2 theta)`
=`((1-sin^2 theta )^2)/(1+ cos^2 theta)+((1-cos^2 theta )^2)/(1+ sin^2 theta)`
=`((1-sin^2 theta )^2 (1+sin^2 )+ (1- cos^2 theta)^2 (1+ cos^2 theta))/((1+ sin^2 theta )( 1+ cos^2 theta))`
=`(cos^4 theta (1+sin^2 theta )+ sin^4 theta (1+cos^2theta))/(1+ sin^2 theta + cos^2 theta + sin^2 theta cos ^2 theta )`
=`(cos^4 theta cos^4 theta sin^2 theta+ sin^4 theta + sin^4 theta cos ^2 theta )/(1+1 sin^2 theta cos^2 theta)`
=`(cos^4 theta + sin^4 theta + sin^2 theta cos^2 theta (sin^2 theta + cos^2 theta))/(2+ sin^2 theta cos^2 theta)`
=`((cos^2 theta)^2 + ( sin^2 theta )^2 + sin^2 theta cos^2 theta (1))/(2+ sin^2 theta cos^2 theta)`
=`((cos^2 theta + sin^2 theta )^2 -2 sin ^2 theta cos^2 theta + sin^2 theta cos^2 theta (1))/(2 + sin^2 theta cos^2 theta)`
=`(1^2+ cos^2 theta sin^2 theta -2 cos^2 theta sin^2 theta)/(2+ sin^2 theta cos^2 theta)`
=`(1-cos^2 theta sin^2 theta)/(2+ sin^2 theta cos^2 theta)`
=RHS
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities
(1 + cot2 A) sin2 A = 1
Prove the following trigonometric identity.
`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`
Prove the following trigonometric identities.
`(tan A + tan B)/(cot A + cot B) = tan A tan B`
Prove that
`sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ)) = 2 sec θ`
If x cos A + y sin A = m and x sin A – y cos A = n, then prove that : x2 + y2 = m2 + n2
Prove the following identities:
`cosecA - cotA = sinA/(1 + cosA)`
`(cos^3 theta +sin^3 theta)/(cos theta + sin theta) + (cos ^3 theta - sin^3 theta)/(cos theta - sin theta) = 2`
If `(x/a sin a - y/b cos theta) = 1 and (x/a cos theta + y/b sin theta ) =1, " prove that "(x^2/a^2 + y^2/b^2 ) =2`
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`
If `sec theta + tan theta = x," find the value of " sec theta`
\[\frac{1 + \tan^2 A}{1 + \cot^2 A}\]is equal to
Prove the following Identities :
`(cosecA)/(cotA+tanA)=cosA`
If x = r sinA cosB , y = r sinA sinB and z = r cosA , prove that `x^2 + y^2 + z^2 = r^2`
Without using trigonometric identity , show that :
`tan10^circ tan20^circ tan30^circ tan70^circ tan80^circ = 1/sqrt(3)`
Prove that sec2 (90° - θ) + tan2 (90° - θ) = 1 + 2 cot2 θ.
If tan α = n tan β, sin α = m sin β, prove that cos2 α = `(m^2 - 1)/(n^2 - 1)`.
Choose the correct alternative:
`(1 + cot^2"A")/(1 + tan^2"A")` = ?
Prove that `(sin^2theta)/(cos theta) + cos theta` = sec θ
Let x1, x2, x3 be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan–1(x + 1) where x1 < x2 < x3 then 2x1 + x2 + x32 is equal to ______.
Prove that (sec θ + tan θ) (1 – sin θ) = cos θ
