Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`
Advertisements
उत्तर
In the given question, we need to prove `((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`
Taking `sin theta` common from the numerator and the denominator of the L.H.S, we get
`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (((sin theta)(cosec theta + 1 -cot theta))/((sin theta)(cosec theta + 1 + cot theta)))^2`
`= ((1 + cosec theta - cot theta)/(1 + cosec theta + cot theta))^2`
Now, using the property `1 + cot^2 theta = cosec^2 theta` we get
`((1 + cosec theta - cot theta)/(1 + cosec theta + cot theta))^2 = (((cosec^2 theta - cot^2 theta) +cosec theta - cot theta)/(1 + cosec theta + cot theta))^2`
Using `a^2 - b^2 = (a + b)(a - b) we get
`(((cosec^2 theta - cot^2 theta)(cosec theta - cot theta))/(1 + cosec theta + cot theta))^2 = (((cosec theta - cot theta)(cosec theta + cot theta + 1))/(1 + cosec theta + cot theta))^2`
`= (cosec theta - cot theta)^2`
Using `cot theta = cos theta/sin theta` and `cosec = 1/sin theta` we get
`(cosec theta - cot theta)^2 = (1/sin theta - cos theta/sin theta)^2`
`= ((1 - cos theta)/sin theta)^2`
Now, using the property `sin^2 theta + cos^2 theta = 1` we get
`(1 - cos theta)^2/sin^2 theta = (1 - cos theta)/(1 - cos^2 theta)`
`= (1 - cos theta)^2/((1 + cos theta)(1 - cos theta))`
`= (1 - cos theta)/(1 + cos theta)`
Hence proved.
APPEARS IN
संबंधित प्रश्न
If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p2 – 1) = 2p
Show that `sqrt((1+cosA)/(1-cosA)) = cosec A + cot A`
Prove the following trigonometric identities.
sin2 A cot2 A + cos2 A tan2 A = 1
Prove the following trigonometric identities.
`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`
Prove the following trigonometric identities.
if `T_n = sin^n theta + cos^n theta`, prove that `(T_3 - T_5)/T_1 = (T_5 - T_7)/T_3`
Prove that
`sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ)) = 2 sec θ`
Prove the following identities:
(cosec A + sin A) (cosec A – sin A) = cot2 A + cos2 A
Prove the following identities:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Prove the following identities:
`(sinA + cosA)/(sinA - cosA) + (sinA - cosA)/(sinA + cosA) = 2/(2sin^2A - 1)`
Prove the following identities:
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (1 + cosA)/sinA`
Prove the following identities:
`(1+ sin A)/(cosec A - cot A) - (1 - sin A)/(cosec A + cot A) = 2(1 + cot A)`
Prove that:
(cosec A – sin A) (sec A – cos A) sec2 A = tan A
If a cos `theta + b sin theta = m and a sin theta - b cos theta = n , "prove that "( m^2 + n^2 ) = ( a^2 + b^2 )`
Prove that:
`"tanθ"/("secθ" – 1) = (tanθ + secθ + 1)/(tanθ + secθ - 1)`
Prove that:
(cosec θ - sinθ )(secθ - cosθ ) ( tanθ +cot θ) =1
Prove the following identity :
`sin^4A + cos^4A = 1 - 2sin^2Acos^2A`
Prove the following identity :
`[1/((sec^2θ - cos^2θ)) + 1/((cosec^2θ - sin^2θ))](sin^2θcos^2θ) = (1 - sin^2θcos^2θ)/(2 + sin^2θcos^2θ)`
Without using trigonometric identity , show that :
`sin(50^circ + θ) - cos(40^circ - θ) = 0`
If A + B = 90°, show that `(sin B + cos A)/sin A = 2tan B + tan A.`
Prove that `cot^2 "A" [(sec "A" - 1)/(1 + sin "A")] + sec^2 "A" [(sin"A" - 1)/(1 + sec"A")]` = 0
