Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(1/(sec^2 theta - cos 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
In the given question, we need to prove
`(1/(sec^2 theta - cos 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)`
Now using `sec theta = 1/ cos theta` and `cosec theta = 1/sin theta` in LHS we get
LHS =`(1/((1/cos^2 theta) - cos^2 theta) + 1/((1/sin^2 theta) - sin^2 theta)) sin^2 theta cos^2 theta`
`= (1/((1 - cos^4 theta)/cos^2 theta) + 1/((1 - sin^4 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`
Further using the identity `a^2 - b^2 = (a + b)(a- b)` we get
LHS = `(cos^2 theta/((1 - cos^2 theta)(1 + cos^2 theta)) + sin^2 theta/((1 - sin^2 theta) (1 + sin^2 theta)))sin^2 theta cos^2 theta`
`= ((cos^2 theta)/(sin^2 theta(1 + cos^2 theta)) + sin^2 theta/(cos^2 theta(1 + sin^2 theta))) sin^2 theta cos^2 theta`
`= ((cos^2 theta(cos^2 theta(1 + sin^2 theta))+sin^2 theta(sin^2 theta(1 + cos^2 theta)))/(sin^2 theta cos^2 theta(1 + cos^2 theta)(1 +sin^2 theta))) sin^2 theta cos^2 theta`
`= ((cos^4 theta(1 + sin^2 theta) + sin^4 theta(1 + cos^2 theta))/((1 + cos^2 theta)(1 + sin^2 theta)))`
Further using the identity `sin^2 theta + cos^2 theta = 1` we get
LHS = `((cos^4 theta + cos^4 theta sin^2 theta + sin^4 theta + sin^4 theta cos^2 theta)/(1 + cos^2 theta + sin^2 theta + sin^2 theta cos^2 theta))`
`= (cos^4 theta + sin^4 theta + cos^2 theta sin^2 theta (cos^2 theta + sin^2 theta)) /(2 + sin^2 theta cos^2theta)`
`= ((cos^4 theta +sin^4 theta +cos^2 theta sin^2theta (1))/(2 + sin^2 theta cos^2 theta))`
Now, from the identity `a^2 + b^2 = (a + b)^2 - 2ab` we get
So,
LHS = `(((cos^2 theta + sin^2 theta)^2 - 2cos^2 theta sin^2 theta +cos^2 theta sin^2 theta)/(2 + sin^2 theta cos^2 theta))`
`= (((1)^2 - cos^2 theta sin^2 theta)/(22 +sin^2 theta cos^2 theta))`
`= ((1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta cos^2 theta))`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that: `(1 – sinθ + cosθ)^2 = 2(1 + cosθ)(1 – sinθ)`
As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.
Prove the following trigonometric identities.
`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`
Prove the following identities:
`(sinA - cosA + 1)/(sinA + cosA - 1) = cosA/(1 - sinA)`
Prove that:
(cosec A – sin A) (sec A – cos A) sec2 A = tan A
`1/((1+ sin θ)) + 1/((1 - sin θ)) = 2 sec^2 θ`
`(sec theta -1 )/( sec theta +1) = ( sin ^2 theta)/( (1+ cos theta )^2)`
`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta )) = 2 cosec theta`
Prove that:
`"tanθ"/("secθ" – 1) = (tanθ + secθ + 1)/(tanθ + secθ - 1)`
Simplify : 2 sin30 + 3 tan45.
What is the value of \[\frac{\tan^2 \theta - \sec^2 \theta}{\cot^2 \theta - {cosec}^2 \theta}\]
Write True' or False' and justify your answer the following :
The value of \[\cos^2 23 - \sin^2 67\] is positive .
\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to
If x = a cos θ and y = b sin θ, then b2x2 + a2y2 =
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Find the value of `θ(0^circ < θ < 90^circ)` if :
`cos 63^circ sec(90^circ - θ) = 1`
Prove that cosec2 (90° - θ) + cot2 (90° - θ) = 1 + 2 tan2 θ.
If 1 + sin2θ = 3 sin θ cos θ, then prove that tan θ = 1 or `1/2`.
Show that: `tan "A"/(1 + tan^2 "A")^2 + cot "A"/(1 + cot^2 "A")^2 = sin"A" xx cos"A"`
`1/sin^2θ - 1/cos^2θ - 1/tan^2θ - 1/cot^2θ - 1/sec^2θ - 1/("cosec"^2θ) = -3`, then find the value of θ.
