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)`
Advertisements
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
Express the ratios cos A, tan A and sec A in terms of sin A.
Prove the following trigonometric identities.
`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`
Prove the following trigonometric identities.
tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B
Prove the following identities:
sec2A + cosec2A = sec2A . cosec2A
Prove the following identities:
`(cosecA - 1)/(cosecA + 1) = (cosA/(1 + sinA))^2`
If sin A + cos A = m and sec A + cosec A = n, show that : n (m2 – 1) = 2 m
Prove that:
(tan A + cot A) (cosec A – sin A) (sec A – cos A) = 1
`tan theta /((1 - cot theta )) + cot theta /((1 - tan theta)) = (1+ sec theta cosec theta)`
`(sec theta + tan theta )/( sec theta - tan theta ) = ( sec theta + tan theta )^2 = 1+2 tan^2 theta + 25 sec theta tan theta `
`(sin theta)/((sec theta + tan theta -1)) + cos theta/((cosec theta + cot theta -1))=1`
If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ?
\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to
Prove the following identity :
`(cosecA - sinA)(secA - cosA)(tanA + cotA) = 1`
Prove the following identity :
`(cos^3θ + sin^3θ)/(cosθ + sinθ) + (cos^3θ - sin^3θ)/(cosθ - sinθ) = 2`
Without using trigonometric table , evaluate :
`cosec49°cos41° + (tan31°)/(cot59°)`
Find the value of `θ(0^circ < θ < 90^circ)` if :
`cos 63^circ sec(90^circ - θ) = 1`
Prove that :(sinθ+cosecθ)2+(cosθ+ secθ)2 = 7 + tan2 θ+cot2 θ.
`sqrt((1 - cos^2theta) sec^2 theta) = tan theta`
If 2sin2θ – cos2θ = 2, then find the value of θ.
Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos (α - β)/2` is ______.
