Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos A`
Advertisements
उत्तर
We need to prove `cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos `
Solving the L.H.S, we get
`cos A/(1 - tan A) + sin A/(1 - cot A)`
= `cos A/(1 - sin A/cos A) + sin A/(1 - cos A/sin A)`
`= cos A/((cos A - sin A)/cos A) + sin A/((sin A - cos A)/sin A)`
`= cos^2 A/(cos A - sin A) + (sin^2 A)/(sin A - cos A)`
`= (cos^2 A - sin^2 A)/(cos A - sin A)`
`= ((cos A + sin A)(cos A - sin A))/(cos A - sin A)` [using `a^2 - b^2 = (a + b)(a -b)`]
= cos A + sin A
= RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
If secθ + tanθ = p, show that `(p^{2}-1)/(p^{2}+1)=\sin \theta`
Without using trigonometric tables evaluate
`(sin 35^@ cos 55^@ + cos 35^@ sin 55^@)/(cosec^2 10^@ - tan^2 80^@)`
Prove the following trigonometric identities.
`(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`
Prove the following identities:
`1/(tan A + cot A) = cos A sin A`
Prove the following identities:
(sec A – cos A) (sec A + cos A) = sin2 A + tan2 A
Prove the following identities:
(cos A + sin A)2 + (cos A – sin A)2 = 2
Prove the following identities:
`(1 + sinA)/cosA + cosA/(1 + sinA) = 2secA`
Prove the following identities:
`cosecA - cotA = sinA/(1 + cosA)`
`(sec^2 theta -1)(cosec^2 theta - 1)=1`
`cos^2 theta /((1 tan theta))+ sin ^3 theta/((sin theta - cos theta))=(1+sin theta cos theta)`
`(cos theta cosec theta - sin theta sec theta )/(costheta + sin theta) = cosec theta - sec theta`
Write the value of `sin theta cos ( 90° - theta )+ cos theta sin ( 90° - theta )`.
If `cos B = 3/5 and (A + B) =- 90° ,`find the value of sin A.
Write the value of tan1° tan 2° ........ tan 89° .
`If sin theta = cos( theta - 45° ),where theta " is acute, find the value of "theta` .
If 5x = sec ` theta and 5/x = tan theta , " find the value of 5 "( x^2 - 1/( x^2))`
Write the value of sin A cos (90° − A) + cos A sin (90° − A).
Write the value of \[\cot^2 \theta - \frac{1}{\sin^2 \theta}\]
If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ?
Write True' or False' and justify your answer the following :
The value of \[\cos^2 23 - \sin^2 67\] is positive .
The value of sin ( \[{45}^° + \theta) - \cos ( {45}^°- \theta)\] is equal to
Prove the following identity :
( 1 + cotθ - cosecθ) ( 1 + tanθ + secθ)
Prove the following identity:
`cosA/(1 + sinA) = secA - tanA`
If m = a secA + b tanA and n = a tanA + b secA , prove that m2 - n2 = a2 - b2
Without using trigonometric identity , show that :
`sec70^circ sin20^circ - cos20^circ cosec70^circ = 0`
Prove that `sin^2 θ/ cos^2 θ + cos^2 θ/sin^2 θ = 1/(sin^2 θ. cos^2 θ) - 2`.
If `cos theta/(1 + sin theta) = 1/"a"`, then prove that `("a"^2 - 1)/("a"^2 + 1)` = sin θ
If tan θ = `9/40`, complete the activity to find the value of sec θ.
Activity:
sec2θ = 1 + `square` ......[Fundamental trigonometric identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square`
sec θ = `square`
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ
If 2sin2β − cos2β = 2, then β is ______.
