Advertisements
Advertisements
प्रश्न
Prove that
`sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ)) = 2 sec θ`
Advertisements
उत्तर
`"LHS" = sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ))`
Taking L.H.S and rationalizing the numerator and denominator with its respective conjugates, we get,
`"LHS" = sqrt((1 + sin θ)/(1 - sin θ) × (1 + sin θ)/(1 + sin θ)) + sqrt((1 - sin θ)/(1 + sin θ) × (1 - sin θ)/(1 - sin θ))`
`"LHS" = sqrt((1 + sin θ)^2/(1 - sin^2 θ)) + sqrt((1 - sin θ)^2/(1 - sin^2 θ))`
`"LHS" = sqrt((1 + sin^2θ)/(1 - sin^2 θ)) + sqrt((1 - sin^2θ)/(1 - sin^2 θ))`
`"LHS" = sqrt((1 + sin^2θ)/(cos^2 θ)) + sqrt((1 - sin^2θ)/(cos^2 θ))`
`"LHS" = (1 + sin θ)/(cos θ) + (1 - sin θ)/(cos θ)`
`"LHS" = (1 + cancel(sin θ) + 1 -cancel(sin θ))/(cos θ)`
LHS = `2/(cos θ)`
LHS = 2. `1/(cos θ)`
LHS = 2. sec θ
RHS = 2. sec θ
LHS = RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
If secθ + tanθ = p, show that `(p^{2}-1)/(p^{2}+1)=\sin \theta`
Prove the following trigonometric identities
`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities.
`tan theta - cot theta = (2 sin^2 theta - 1)/(sin theta cos theta)`
Prove the following identities:
`cosA/(1 - sinA) = sec A + tan A`
If ` cot A= 4/3 and (A+ B) = 90° ` ,what is the value of tan B?
What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?
Write True' or False' and justify your answer the following:
\[ \cos \theta = \frac{a^2 + b^2}{2ab}\]where a and b are two distinct numbers such that ab > 0.
Prove the following identity :
`cos^4A - sin^4A = 2cos^2A - 1`
Find the value of `θ(0^circ < θ < 90^circ)` if :
`tan35^circ cot(90^circ - θ) = 1`
Find x , if `cos(2x - 6) = cos^2 30^circ - cos^2 60^circ`
Find the value of ( sin2 33° + sin2 57°).
Prove that `sqrt((1 + sin A)/(1 - sin A))` = sec A + tan A.
Prove that `( tan A + sec A - 1)/(tan A - sec A + 1) = (1 + sin A)/cos A`.
Prove that `(sin θ. cos (90° - θ) cos θ)/sin( 90° - θ) + (cos θ sin (90° - θ) sin θ)/(cos(90° - θ)) = 1`.
If A + B = 90°, show that sec2 A + sec2 B = sec2 A. sec2 B.
Choose the correct alternative:
sin θ = `1/2`, then θ = ?
`sqrt((1 - cos^2theta) sec^2 theta) = tan theta`
If 2sin2θ – cos2θ = 2, then find the value of θ.
Prove that (sec θ + tan θ) (1 – sin θ) = cos θ
Factorize: sin3θ + cos3θ
Hence, prove the following identity:
`(sin^3θ + cos^3θ)/(sin θ + cos θ) + sin θ cos θ = 1`
