Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
Advertisements
उत्तर
We have prove that
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
We know that `sin^2 A + cos^2 A = 1`
So,
(2 + cot A + tan A)(sin A - cos A)
`= (1 + cos A/sin A + sin A/cos A)(sin A - cos A)`
`= ((sin A cos A + cos^2 A + sin^2 A)/(sin A cos A)) (sin A - cos A)`
`= ((sin A cos A + 1)/(sin A cos A))(sin A - cos A)`
`= ((sin A - cos A)(sin A cos A + 1))/(sin A cos A)`
`= (sin^2 A cos A + sin A - cos^2 A sin A - cos A)/(sin A cos A)`
`= ((sin^2 A cos A - cos A) + (sin A - cos^2 A sin A))/(sin A cos A)`
`= (cos A (sin^2 A - 1)+ sin A (1 - sin^2 A))/(sin A cos A)`
`= (cos A (-cos^2 A) + sin A (sin^2 A))/(sin A cos A)`
`= (-cos^3 A + sin^3 A)/(sin A cos A)`
`= (sin^3 A - cos^3 A)/(sin A cos A)`
`= (sin^2 A)/cos A - cos^2 A/sin A`
`= sin A/cos A sin A - cos A/sin A cos A`
`= tan A sin A - cot A cos A`
= sin A tan A - cos A cot A
Now
`sec A/(cosec^2 A) - (cosec A)/sec^2 A = (1/cos A)/(1/sin^2 A) - (1/sin A)/(1/cos^2 A)`
`= sin^2 A/cos A - cos^2 A/sin A`
`= sin A sin A/cos a - cos A cos A/sin A`
= sin A tan A - cos A cot A
Hence proved.
APPEARS IN
संबंधित प्रश्न
If `sec alpha=2/sqrt3` , then find the value of `(1-cosecalpha)/(1+cosecalpha)` where α is in IV quadrant.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
`(sintheta - 2sin^3theta)/(2costheta - costheta) =tan theta`
If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1
Prove the following identities:
`(sinA - cosA + 1)/(sinA + cosA - 1) = cosA/(1 - sinA)`
`sin theta (1+ tan theta) + cos theta (1+ cot theta) = ( sectheta+ cosec theta)`
If `(cot theta ) = m and ( sec theta - cos theta) = n " prove that " (m^2 n)(2/3) - (mn^2)(2/3)=1`
Write the value of cos1° cos 2°........cos180° .
What is the value of 9cot2 θ − 9cosec2 θ?
If cosec θ = 2x and \[5\left( x^2 - \frac{1}{x^2} \right)\] \[2\left( x^2 - \frac{1}{x^2} \right)\]
If \[sec\theta + tan\theta = x\] then \[tan\theta =\]
Prove the following identity :
`tan^2A - tan^2B = (sin^2A - sin^2B)/(cos^2Acos^2B)`
If x = asecθ + btanθ and y = atanθ + bsecθ , prove that `x^2 - y^2 = a^2 - b^2`
If sinA + cosA = `sqrt(2)` , prove that sinAcosA = `1/2`
Find the value of x , if `cosx = cos60^circ cos30^circ - sin60^circ sin30^circ`
Without using trigonometric identity , show that :
`sin(50^circ + θ) - cos(40^circ - θ) = 0`
Evaluate:
`(tan 65^circ)/(cot 25^circ)`
Prove that sec2 (90° - θ) + tan2 (90° - θ) = 1 + 2 cot2 θ.
Prove that: sin6θ + cos6θ = 1 - 3sin2θ cos2θ.
Prove that
sec2A – cosec2A = `(2sin^2"A" - 1)/(sin^2"A"*cos^2"A")`
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 ______.
