Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1
उत्तर
We have to prove `(1 + tan^2 theta)(1 - sin theta)(1 + sin theta) = 1`
We know that
`sin^2 theta + cos^2 theta = 1`
`sec^2 theta - tan^2 theta = 1`
So
`(1 + tan^2 theta)(1 - sin theta) = (1 + tan^2 theta){(1 - sin theta)(1 + sin theta)}`
` = (1 + tan^2 theta)(1 - sin^2 theta)`
`= sec^2 theta cos^2 theta`
` = 1/cos^2 theta cos^2 theta`
= 1
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identity.
`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`
Prove the following trigonometric identities.
(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)
Prove that
`sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ)) = 2 sec θ`
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`
Prove the following identities:
`cot^2A/(cosecA + 1)^2 = (1 - sinA)/(1 + sinA)`
If `(cosec theta - sin theta )= a^3 and (sec theta - cos theta ) = b^3 , " prove that " a^2 b^2 ( a^2+ b^2 ) =1`
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`
Write the value of tan1° tan 2° ........ tan 89° .
If x = a sin θ and y = bcos θ , write the value of`(b^2 x^2 + a^2 y^2)`
If tanθ `= 3/4` then find the value of secθ.
What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?
If a cot θ + b cosec θ = p and b cot θ − a cosec θ = q, then p2 − q2
If a cos θ − b sin θ = c, then a sin θ + b cos θ =
Prove the following identity :
`(cotA - cosecA)^2 = (1 - cosA)/(1 + cosA)`
Proved that cosec2(90° - θ) - tan2 θ = cos2(90° - θ) + cos2 θ.
If `sqrt(3)` sin θ – cos θ = θ, then show that tan 3θ = `(3tan theta - tan^3 theta)/(1 - 3 tan^2 theta)`
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
sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A
Prove that sin6A + cos6A = 1 – 3sin2A . cos2A
If tan θ – sin2θ = cos2θ, then show that sin2 θ = `1/2`.
