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Prove that :(1+cotA +tanA) (sinA+cosA)=secA upon cosec Square A - cosec A upon sec square A =sinA tanA - cot A cos A?
Most Upvoted Answer
Prove that :(1+cotA +tanA) (sinA+cosA)=secA upon cosec Square A - cose...
LHS= (1+cotA+tanA)(sinA-cosA)
=sinA+sinAcotA+sinAtanA-cosA-cosAcotA-
cosAtanA
=sinA+sinAcosA/sinA+sinAsinA/cosA-cosA
-cosAcosA/sinA-cosAsinA/cosA
=sinA+cosA+sin^2A/cosA-cosA-
cos^2A/sinA-sinA
=sin^2A/cosA-cos^2A/sinA
=sinA(sinA/cosA)-cosA(cosA/sinA) ---(1)
=1/cosecA(1/cosecA/1/secA)-1/secA(1/secA/
1/cosecA)
=(1/cosec^2A)(secA)-(1/sec^2)(cosecA)
=secA/cosec^2A-cosecA/sec^2A ---(2)
=sinAtanA-cosAcotA ----(3)
from (1)
from 2 and 3
=RHS
Community Answer
Prove that :(1+cotA +tanA) (sinA+cosA)=secA upon cosec Square A - cose...
Proof:
To prove the given equation:
(1 cotA tanA)(sinA cosA) = secA (cosec^2A - cosecA) sec^2A = sinA tanA - cotA cosA
We will start by simplifying the left-hand side (LHS) of the equation and then simplify the right-hand side (RHS) separately. Finally, we will compare the two sides to check if they are equal.
Simplifying the LHS:
We are given: (1 cotA tanA)(sinA cosA)
Using the trigonometric identities:
cotA = cosA/sinA
tanA = sinA/cosA
We can substitute these values into the equation:
(1 cotA tanA)(sinA cosA) = (1 * cosA/sinA * sinA/cosA)(sinA cosA)
= (cosA/cosA)(sinA/sinA)(sinA cosA)
= 1 * 1 * (sinA cosA)
= sinA cosA
Simplifying the RHS:
We are given: secA (cosec^2A - cosecA) sec^2A
Using the trigonometric identities:
secA = 1/cosA
cosecA = 1/sinA
cosec^2A = (1/sinA)^2 = 1/sin^2A = csc^2A
sec^2A = (1/cosA)^2 = 1/cos^2A
We can substitute these values into the equation:
secA (cosec^2A - cosecA) sec^2A = (1/cosA)(csc^2A - 1/sinA)(1/cos^2A)
= (1/cosA)((1/sin^2A) - (1/sinA))(1/cos^2A)
= (1/cosA)((1 - sinA)/sin^2A)(1/cos^2A)
= (1/cosA)(1 - sinA)(1/cosA)(1/sin^2A)
= (1/cosA)(1 - sinA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/(sinA * cosA))
= (1 - sinA)(1/(sinA * cosA))(1/cosA)
= (1 - sinA)/sinA * 1/cosA
= (1/sinA - sinA/sinA)(1/cosA)
= (cosecA - 1)(1/cosA)
= (1/cosecA - 1/cosA)
= (cosecA/secA - 1
To prove the given equation:
(1 cotA tanA)(sinA cosA) = secA (cosec^2A - cosecA) sec^2A = sinA tanA - cotA cosA
We will start by simplifying the left-hand side (LHS) of the equation and then simplify the right-hand side (RHS) separately. Finally, we will compare the two sides to check if they are equal.
Simplifying the LHS:
We are given: (1 cotA tanA)(sinA cosA)
Using the trigonometric identities:
cotA = cosA/sinA
tanA = sinA/cosA
We can substitute these values into the equation:
(1 cotA tanA)(sinA cosA) = (1 * cosA/sinA * sinA/cosA)(sinA cosA)
= (cosA/cosA)(sinA/sinA)(sinA cosA)
= 1 * 1 * (sinA cosA)
= sinA cosA
Simplifying the RHS:
We are given: secA (cosec^2A - cosecA) sec^2A
Using the trigonometric identities:
secA = 1/cosA
cosecA = 1/sinA
cosec^2A = (1/sinA)^2 = 1/sin^2A = csc^2A
sec^2A = (1/cosA)^2 = 1/cos^2A
We can substitute these values into the equation:
secA (cosec^2A - cosecA) sec^2A = (1/cosA)(csc^2A - 1/sinA)(1/cos^2A)
= (1/cosA)((1/sin^2A) - (1/sinA))(1/cos^2A)
= (1/cosA)((1 - sinA)/sin^2A)(1/cos^2A)
= (1/cosA)(1 - sinA)(1/cosA)(1/sin^2A)
= (1/cosA)(1 - sinA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/cosA)(1/(sinA * sinA))
= (1 - sinA)(1/cosA)(1/(sinA * cosA))
= (1 - sinA)(1/(sinA * cosA))(1/cosA)
= (1 - sinA)/sinA * 1/cosA
= (1/sinA - sinA/sinA)(1/cosA)
= (cosecA - 1)(1/cosA)
= (1/cosecA - 1/cosA)
= (cosecA/secA - 1
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Prove that :(1+cotA +tanA) (sinA+cosA)=secA upon cosec Square A - cosec A upon sec square A =sinA tanA - cot A cos A?
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