Advertisements
Advertisements
Question
Prove the following trigonometric identities.
(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1
Advertisements
Solution
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
RELATED QUESTIONS
The angles of depression of two ships A and B as observed from the top of a light house 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships. Give your answer correct to the nearest whole number.
Prove the following trigonometric identities.
(sec A + tan A − 1) (sec A − tan A + 1) = 2 tan A
Prove the following identities:
cosec4 A – cosec2 A = cot4 A + cot2 A
Prove the following identities:
sec2A + cosec2A = sec2A . cosec2A
Prove the following identities:
(cosec A + sin A) (cosec A – sin A) = cot2 A + cos2 A
Prove the following identities:
sec2 A . cosec2 A = tan2 A + cot2 A + 2
If x = r cos A cos B, y = r cos A sin B and z = r sin A, show that : x2 + y2 + z2 = r2
Prove that:
`(sinA - cosA)(1 + tanA + cotA) = secA/(cosec^2A) - (cosecA)/(sec^2A)`
Prove that:
(tan A + cot A) (cosec A – sin A) (sec A – cos A) = 1
`sin^2 theta + cos^4 theta = cos^2 theta + sin^4 theta`
If x = a sin θ and y = bcos θ , write the value of`(b^2 x^2 + a^2 y^2)`
If `sin theta = x , " write the value of cot "theta .`
Prove that:
`(sin^2θ)/(cosθ) + cosθ = secθ`
Given `cos38^circ sec(90^circ - 2A) = 1` , Find the value of <A
Prove that: sin4 θ + cos4θ = 1 - 2sin2θ cos2 θ.
Prove that `(tan θ + sin θ)/(tan θ - sin θ) = (sec θ + 1)/(sec θ - 1)`
cot θ . tan θ = ?
Prove that `(1 + sin B)/(cos B) + (cos B)/(1 + sin B) = 2 sec B`.
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ.
If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ = `sqrt(a^2 + b^2 - c^2)`.
