Advertisements
Advertisements
Question
Prove the following trigonometric identities
`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`
Advertisements
Solution
We have to prove `(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`
Consider the expression
L.H.S
`(1 + tan^2 theta)/(1 + cot^2 theta) = (1 + tan^2 theta)/(1 + 1/(tan^2 theta))`
= `(1 +tan^2 theta)/((tan^2 theta + 1)/tan^2 theta)`
`= tan^2 theta (1 + tan^2 theta)/(1 + tan^2 theta)`
`= tan^2 theta`
= R.H.S
Again, we have
L.H.S
`((1 - tan theta)/(1 - cot theta))^2 = ((1 - tan theta)/(1 - 1/(tan theta)))^2`
`= ((1 - tan theta)/((tan theta - 1)/tan theta))^2`
`=[(tantheta(1-tantheta))/-(1-tantheta)]^2`
`=(-tantheta)^2=tan^2theta`
= R.H.S
APPEARS IN
RELATED QUESTIONS
Prove that:
sec2θ + cosec2θ = sec2θ x cosec2θ
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1
Prove that:
`(sinA - sinB)/(cosA + cosB) + (cosA - cosB)/(sinA + sinB) = 0`
Prove that:
`1/(cosA + sinA - 1) + 1/(cosA + sinA + 1) = cosecA + secA`
Prove the following identities:
`cosA/(1 + sinA) + tanA = secA`
Prove the following identities:
cosec4 A (1 – cos4 A) – 2 cot2 A = 1
`(sec^2 theta -1)(cosec^2 theta - 1)=1`
`cot^2 theta - 1/(sin^2 theta ) = -1`a
If `sec theta = x ,"write the value of tan" theta`.
\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to
Without using trigonometric identity , show that :
`sec70^circ sin20^circ - cos20^circ cosec70^circ = 0`
If tan θ = 2, where θ is an acute angle, find the value of cos θ.
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
If x sin3θ + y cos3 θ = sin θ cos θ and x sin θ = y cos θ , then show that x2 + y2 = 1.
Prove that `tan A/(1 + tan^2 A)^2 + cot A/(1 + cot^2 A)^2 = sin A.cos A`
Prove that `(sin 70°)/(cos 20°) + (cosec 20°)/(sec 70°) - 2 cos 70° xx cosec 20°` = 0.
cos θ . sec θ = ?
Prove that `(cot A)/(1 - tan A) + (tan A)/(1 - cot A) = 1 + tan A + cot A = sec A . "cosec" A + 1`.
If 2sin2β − cos2β = 2, then β is ______.
