Advertisements
Advertisements
Question
Prove the following identities.
sec4 θ (1 – sin4 θ) – 2 tan2 θ = 1
Solution
L.H.S = sec4 θ (1 – sin4 θ) – 2 tan2 θ
= `1/cos^4 theta [1 - (sin^2 theta)^2]- 2 xx (sin^2 theta)/(cos^2 theta)`
= `1/(cos^4 theta) (1 + sin^2 theta) (1 - sin^2 theta) - 2 (sin^2 theta)/(cos^2 theta)`
= `1/(cos^4 theta) xx cos^2 theta (1 + sin^2 theta) - 2 (sin^2 theta)/(cos^2 theta)`
= `(1 + sin^2 theta)/(cos^2 theta) - (2sin^2 theta)/(cos^2 theta)`
= `(1 + sin^2 theta - 2sin^2 theta)/(cos^2 theta)`
= `(1 - sin^2 theta)/(cos^2 theta)`
= `(cos^2 theta)/(cos^2 theta)`
L.H.S = R.H.S
∴ sec4 θ (1 – sin4 θ) – 2 tan2 θ = 1
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities
`((1 + sin theta)^2 + (1 + sin theta)^2)/(2cos^2 theta) = (1 + sin^2 theta)/(1 - sin^2 theta)`
Prove the following identities:
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (1 + cosA)/sinA`
Prove the following identities:
`((cosecA - cotA)^2 + 1)/(secA(cosecA - cotA)) = 2cotA`
`If sin theta = cos( theta - 45° ),where theta " is acute, find the value of "theta` .
Simplify : 2 sin30 + 3 tan45.
Write True' or False' and justify your answer the following :
The value of sin θ+cos θ is always greater than 1 .
Prove the following identity :
`(1 + cotA + tanA)(sinA - cosA) = secA/(cosec^2A) - (cosecA)/sec^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)`
If x = acosθ , y = bcotθ , prove that `a^2/x^2 - b^2/y^2 = 1.`
Let x1, x2, x3 be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan–1(x + 1) where x1 < x2 < x3 then 2x1 + x2 + x32 is equal to ______.
