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Questions
cos4 A − sin4 A is equal to ______.
(cos4 A − sin4 A) on simplification, gives
Options
2 cos2 A + 1
2 cos2 A − 1
2 sin2 A − 1
2 sin2 A + 1
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Solution
cos4 A − sin4 A is equal to 2 cos2 A − 1.
Explanation:
The given expression is cos4 A − sin4 A.
Factorising the given expression, we have
cos4 A − sin4 A = [(cos2 A)2 − (sin2 A)2]
= (cos2 A + sin2 A) × (cos2 A − sin2 A) ...[∵ (a2 − b2) = (a + b)(a − b)]
= cos2 A − sin2 A ...[∵ sin2 A + cos2 A = 1]
= cos2 A − (1 − sin2 A)
= cos2 A − 1 + cos2 A
= 2 cos2 A − 1
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To prove cot θ + tan θ = cosec θ × sec θ, complete the activity given below.
Activity:
L.H.S = `square`
= `square/sintheta + sintheta/costheta`
= `(cos^2theta + sin^2theta)/square`
= `1/(sintheta*costheta)` ......`[cos^2theta + sin^2theta = square]`
= `1/sintheta xx 1/square`
= `square`
= R.H.S
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If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ = `sqrt(a^2 + b^2 - c^2)`.
Show that, cotθ + tanθ = cosecθ × secθ
Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
sec θ when expressed in term of cot θ, is equal to ______.
(1 – cos2 A) is equal to ______.
