Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities
tan2 A + cot2 A = sec2 A cosec2 A − 2
Advertisements
उत्तर
In the given question, we need to prove tan2 A + cot2 A = sec2 A cosec2 A − 2
Now using `tan theta = sin theta/cos theta` and `cot theta = cos theta/sin theta` in LHS we get
`tan^2 A + cot^2 A = sin^2 A/cos^2 A + cos^2 A/sin^2 A`
`= (sin^4 A + cos^4 A)/(cos^2 A sin^2 A)`
`= ((sin^2 A)^2 + (cos^2 A)^2)/(cos^2 A sin^2 A)`
Further, using the identity `a^2 + b^2 = (a + b)^2 - 2ab` we get
`((sin^2 A)^2 + (cos^2 A)^2)/(cos^2 A sin^2 A) = ((sin^2 A + cos^ A)^2 - 2 sin^2 A cos^2 A)/(sin^2 A cos^2 A)`
`= ((1)^2 - 2sin^2 A cos^2 A)/(sin^2 A cos^2 A)`
`= 1/(sin^2 A cos^2 A) - (2 sin^2 A cos^2 A)/(sin^2 A cos^2 A`
`= cosec^2 A sec^2 A - 2`
Since L.H.S = R.H.S
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that: `(1 – sinθ + cosθ)^2 = 2(1 + cosθ)(1 – sinθ)`
Prove that `\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin\theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2\sin^{2}\theta -1}`
Prove that ` \frac{\sin \theta -\cos \theta +1}{\sin\theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta }` using the identity sec2 θ = 1 + tan2 θ.
`(1+tan^2A)/(1+cot^2A)` = ______.
Prove the following trigonometric identities.
(sec2 θ − 1) (cosec2 θ − 1) = 1
Prove the following trigonometric identities.
`tan theta + 1/tan theta` = sec θ.cosec θ
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
`1/((1+ sin θ)) + 1/((1 - sin θ)) = 2 sec^2 θ`
`(tan theta)/((sec theta -1))+(tan theta)/((sec theta +1)) = 2 sec theta`
If tan A = n tan B and sin A = m sin B , prove that `cos^2 A = ((m^2-1))/((n^2 - 1))`
Write the value of `(1 - cos^2 theta ) cosec^2 theta`.
If \[\sin \theta = \frac{1}{3}\] then find the value of 2cot2 θ + 2.
Prove the following identity :
`(cosA + sinA)^2 + (cosA - sinA)^2 = 2`
Prove the following identities:
`(tan"A"+tan"B")/(cot"A"+cot"B")=tan"A"tan"B"`
If tan θ = 2, where θ is an acute angle, find the value of cos θ.
Verify that the points A(–2, 2), B(2, 2) and C(2, 7) are the vertices of a right-angled triangle.
If x = a sec θ + b tan θ and y = a tan θ + b sec θ prove that x2 - y2 = a2 - b2.
Prove the following identities.
`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ
Prove the following identities.
`(sin "A" - sin "B")/(cos "A" + cos "B") + (cos "A" - cos "B")/(sin "A" + sin "B")`
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 ______.
