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Math hw
5- (cos2θ/ sinθ) + sinθ = cotθ/ secθ
LHS: Left Hand Side
RHS: Right Hand Side
LHS= (Cos2θ/ sinθ) + sinθ
Cos2θ = cos2θ-sin2θ
= ((cos2θ – sin2θ)/ sinθ) + sinθ
= (cos2θ – sin2θ + sin2θ)/ sinθ
= cos2θ/ sinθ
= (cosθ/ sinθ)* cosθ
cosθ/ sinθ=cotθ and cosθ=1/ secθ
= cotθ/ secθ = RHS
Hence proved LHS = RHS
6- tanθ + cotθ = 2/ sin2θ
LHS= tanθ + cotθ
tanθ = sinθ/ cosθ and cotθ = cosθ/ sinθ
= sinθ/ cosθ + cosθ/ sinθ
= (sin2θ + cos2θ)/ sinθcosθ
sin2θ + cos2θ = 1
= 1/ sinθcosθ
Multiply and divide by 2
=2/ 2sinθcosθ
sin2θ = 2sinθcosθ
=2/ sin2θ = RHS
Hence proved LHS = RHS
7- sin2θ = 2tanθ/ (1 + tan2θ)
RHS= 2tanθ/ (1 + tan2θ)
tanθ = sinθ/ cosθ
= 2 (sinθ/ cosθ)/ (1 + sin2θ/ cos2θ)
= 2 (sinθ/ cosθ)/ ((cos2θ + sin2θ)/ cos2θ)
sin2θ + cos2θ = 1
= 2 (sinθ/ cosθ)/ (1/ cos2θ)
=2 (sinθcos2θ / cosθ)
=2sinθcosθ
sin2θ = 2sinθcosθ
= sin2θ = LHS
Hence proved LHS = RHS
8- cos2θ = (1 - tan2θ)/ (1 + tan2θ)
RHS= (1 - tan2θ)/ (1 + tan2θ)
tanθ = sinθ/ cosθ
= (1 - (sin2θ/ cos2θ))/ (1 + (sin2θ/ cos2θ))
= ((cos2θ - sin2θ)/ cos2θ)/ ((cos2θ + sin2θ)/ cos2θ)
cos2θ + sin2θ = 1 & cos2θ - sin2θ= cos2θ
= (cos2θ/ cos2θ)/ (1/ cos2θ)
= cos2θ = LHS
Hence proved LHS = RHS
9- sin2θsec2θ = 2tanθ
LHS= sin2θsec2θ
sin2θ= 2sinθcosθ and sec2θ= 1+ tan2θ
= 2sinθcosθ (1+ tan2θ)
tanθ= sinθ/ cosθ
= 2sinθcosθ + 2sinθcosθ (sin2θ/ cos2θ)
= (2sinθcos3θ + 2 sin3θcosθ)/ cos2θ
= (2sinθcosθ (cos2θ + sin2θ))/ cos2θ
cos2θ + sin2θ = 1
= 2sinθcosθ/ cos2θ
= 2sinθ/ cosθ
tanθ= sinθ/ cosθ
=2tanθ = RHS
Hence proved LHS = RHS
10- 2 - sec2θ = cos2θsec2θ
LHS= 2 - sec2θ
sec2θ = 1 + tan2θ
= 2 – (1 + tan2θ)
= 1 – tan2θ
tanθ = sinθ/ cosθ
= 1 – (sin2θ/ cos2θ)
= (cos2θ - sin2θ)/ cos2θ
cos2θ - sin2θ= cos2θ and sec2θ= 1/ cos2θ
= cos2θsec2θ = RHS
Hence proved LHS = RHS
11- (cosθ – sinθ)2 = 1 - sin2θ
LHS= (cosθ – sinθ)2
= cos2θ + sin2θ – 2sinθcosθ
cos2θ + sin2θ= 1 and sin2θ = 2sinθcosθ
= 1 - sin2θ = RHS
Hence proved LHS = RHS
12- (cosθ + sinθ)2 = 1 + sin2θ
LHS= (cosθ + sinθ)2
= cos2θ + sin2θ + 2sinθcosθ
cos2θ + sin2θ= 1 and sin2θ= 2sinθcosθ
= 1 + sin2θ = RHS
Hence proved LHS = RHS
13- (2sin2θ/ sin2θ) + cotθ = secθcscθ
LHS= (2sin2θ/ sin2θ) + cotθ
sin2θ = 2sinθcosθ and cotθ = cosθ/ sinθ
= (2sin2θ/ 2sinθcosθ) + cosθ/ sinθ
= (sinθ/ cosθ) + (cosθ/ sinθ)
= (sin2θ + cos2θ)/ cosθsinθ
sin2θ + cos2θ = 1
= 1/ cosθsinθ
secθ = 1/ cosθ and cscθ = 1/ sinθ
= secθcscθ = RHS
Hence proved LHS = RHS
14- cos2θ = (cot2θ -1)/ (cot2θ + 1)
RHS= (cot2θ -1)/ (cot2θ + 1)
cotθ = cosθ/ sinθ
= ((cos2θ/ sin2θ) – 1)/ ((cos2θ/ sin2θ) + 1)
= ((cos2θ - sin2θ)/ sin2θ)/ ((cos2θ + sin2θ)/ sin2θ)
cos2θ =cos2θ - sin2θ and cos2θ + sin2θ= 1
= cos2θ = LHS
Hence proved LHS = RHS
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