Advertisements
Advertisements
Question
Prove that:
`"tan A"/(1 + "tan"^2 "A")^2 + "Cot A"/(1 + "Cot"^2 "A")^2 = "sin A cos A"`.
Advertisements
Solution
`"tan A"/(1 + "tan"^2 "A")^2 + "cot A"/(1 + "cot"^2 "A")^2 = "sin A cos A"`.
LHS = `"tan A"/(1 + "tan"^2 "A")^2 + "cot A"/(1 + "cot"^2 "A")^2`
LHS = `"tan A"/("sec"^2 "A")^2 + "cot A"/("cosec"^2 "A")^2 ...{( 1 + "tan"^2θ = "sec"^2θ),(1 + "cot"^2θ = "cosec"^2θ):}`
LHS = `"tan A" × 1/("sec"^2 "A")^2 + "cot A" × 1/("cosec"^2 "A")^2`
LHS = `"sin A"/"cos A" × "cos"^4 "A" + "cos A"/"sin A" × "sin"^4 "A" ...{(cosθ = 1/sec θ),(sin θ = 1/"cosecθ"):}`
LHS = sinA cos3A + cosA sin3A
LHS = sinA cosA (cos2A + sin2A)
LHS = sinA cosA.(1) ...(cos2A + sin2A = 1)
LHS = sinA cosA
RHS = sinA cosA
LHS = RHS
Hence proved.
APPEARS IN
RELATED QUESTIONS
If acosθ – bsinθ = c, prove that asinθ + bcosθ = `\pm \sqrt{a^{2}+b^{2}-c^{2}`
Prove the following trigonometric identities:
`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`
Prove the following identities:
sec2A + cosec2A = sec2A . cosec2A
Prove the following identities:
`(1+ sin A)/(cosec A - cot A) - (1 - sin A)/(cosec A + cot A) = 2(1 + cot A)`
If sin A + cos A = m and sec A + cosec A = n, show that : n (m2 – 1) = 2 m
`cos^2 theta + 1/((1+ cot^2 theta )) =1`
`sin^6 theta + cos^6 theta =1 -3 sin^2 theta cos^2 theta`
`(sin theta +cos theta )/(sin theta - cos theta)+(sin theta- cos theta)/(sin theta + cos theta) = 2/((sin^2 theta - cos ^2 theta)) = 2/((2 sin^2 theta -1))`
If `sin theta = x , " write the value of cot "theta .`
Prove that secθ + tanθ =`(costheta)/(1-sintheta)`.
Write True' or False' and justify your answer the following :
The value of the expression \[\sin {80}^° - \cos {80}^°\]
If sec θ + tan θ = x, then sec θ =
If a cos θ + b sin θ = m and a sin θ − b cos θ = n, then a2 + b2 =
If a cos θ − b sin θ = c, then a sin θ + b cos θ =
Prove the following identity :
`(secθ - tanθ)^2 = (1 - sinθ)/(1 + sinθ)`
If m = a secA + b tanA and n = a tanA + b secA , prove that m2 - n2 = a2 - b2
Without using trigonometric identity , show that :
`sec70^circ sin20^circ - cos20^circ cosec70^circ = 0`
Prove the following identities: sec2 θ + cosec2 θ = sec2 θ cosec2 θ.
Prove the following identities.
sec4 θ (1 – sin4 θ) – 2 tan2 θ = 1
If 2 cos θ + sin θ = `1(θ ≠ π/2)`, then 7 cos θ + 6 sin θ is equal to ______.
