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Math15- cos2θsinθ+sinθ=cscθ-sinθ

LHS: Left Hand Side

RHS: Right Hand Side

LHS= cos2θsinθ+sinθcos2θ=cos2θ-sin2θ

=cos2θ-sin2θsinθ+sinθLCD: sinθ sinθ

= cos2θ-sin2θ+sin2θsinθ where-sin2θ+sin2θ=0

cos2θ = 1 - sin2θ

=1-sin2θsinθ

=1sinθ- sin2θsinθ

cscθ = 1/ sinθ

= cscθ – sinθ = RHS

Hence proved LHS = RHS

16- 2tanθ-sin2θ2sin2θ=tanθ

LHS=2tanθ-sin2θ2sin2θ

sin2θ=(1-cos2θ)

=2sinθcosθ-2sinθcosθ2(1-cos2θ)

=2(sinθ-sinθcos2θcosθ) 2(1-cos2θ)

=sinθ1-cos2θcosθ(1-cos2θ)

= tanθ = RHS

Hence proved LHS = RHS

17- 2cosθ - cos2θcosθ=secθLHS= 2cosθ - cos2θcosθ

Cos2θ = Cos2θ – sin2θ LCD: cosθcosθ

= 2cos2θ-(cos2θ-sin2θ)cosθ

= 2cos2θ-cos2θ+sin2θcosθ

= cos2θ+sin2θcosθ

Cos2θ + sin2θ = 1

= 1cosθ

secθ=1cosθ

= secθ = RHS

Hence proved LHS = RHS

18- cos2θ+ cosθ+1sin2θ+sinθ=cotθ

LHS=cos2θ+ cosθ+1sin2θ+sinθ

Cos2θ = Cos2θ – sin2θ and sin2θ =2sinθcosθ= cos2θ-sin2θ+ cosθ+12sinθcosθ+sinθ

sin2θ=1-cos2θ

=cos2θ-(1-cos2θ)+ cosθ+12sinθcosθ+sinθ

=cos2θ-1+cos2θ+ cosθ+12sinθcosθ+sinθ

=2cos2θ+ cosθ2sinθ(1+cosθ)

=2cosθ(cos+ 1)2sinθ(cosθ+1)

=cosθsinθ

=cotθ = RHS

Hence proved LHS = RHS

19- sinθ+sin2θsecθ+2=sinθcosθ

LHS=sinθ+sin2θsecθ+2

Sin2θ = 2sinθcosθsecθ=1cosθ

=sinθ+2sinθcosθ1cosθ+2

=sinθ+2sinθcosθ1+2cosθcosθ, LCD=1cosθ/cosθ

=sinθ(1+2cosθ)1+2cosθcosθ

=sinθ1cosθ

=sinθcosθ = RHS

Hence proved LHS=RHS

20- 1+cos2θ1-cos2θ=cot2θ

LHS= 1+cos2θ1-cos2θ

Cos2θ = Cos2θ – sin2θ

= 1+cos2θ –sin2θ1-(cos2θ –sin2θ)

= 1+cos2θ –sin2θ1-cos2θ+sin2θ

cos2θ = 1 - sin2θ

sin2θ = 1 - cos2θ

= 1+cos2θ –(1- cos2θ)1-(1-sin2θ)+sin2θ

= 2cos2θ 2sin2θ

=cot2θ = RHS

Hence proved LHS = RHS

21- sinπ6-θ+sinπ6+θ=cosθ

LHS=sinπ6-θ+sinπ6+θ

sin ( a + b) = sin(a)cos(b) + cos(a)sin(b)

sin ( a - b) = sin(a)cos(b) - cos(a)sin(b)

=sinπ6cosθ-cosπ6sinθ+sinπ6cosθ+cosπ6sinθ

=2sinπ6cosθ

sinπ6= 12

=22cosθ

= cosθ = RHS

Hence proved LHS = RHS

22- cosπ4-θ-cosπ4+θ=√2sinθ

cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

LHS= cosπ4-θ-cosπ4+θ

=cosπ4cosθ+sinπ4sinθ -(cosπ4cosθ-sinπ4sinθ)

=cosπ4cosθ+sinπ4sinθ -cosπ4cosθ+sinπ4sinθ=2sinπ4sinθ

sinπ4=√22=2√22sinθ = √2sinθ = RHS

Hence proved LHS = RHS

23- sinπ2+θ-sinπ2-θ=0

LHS= sinπ2+θ-sinπ2-θ

sin ( a + b) = sin(a)cos(b) + cos(a)sin(b)

sin ( a - b) = sin(a)cos(b) - cos(a)sin(b)

=sinπ2cosθ+cosπ2sinθ-(sinπ2cosθ-cosπ2sinθ)

=sinπ2cosθ+cosπ2sinθ-sinπ2cosθ+cosπ2sinθ

=cosπ2sinθ

cosπ2=0

= 0 = RHS

Hence proved LHS = RHS

24- sinπ4+θ+cosπ4+θ=√2cosθ

LHS= sinπ4+θ+cosπ4+θ

Sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

=sinπ4cosθ+cosπ4sinθ+ cosπ4cosθ-sinπ4sinθ=√22cosθ+ √22sinθ+√22cos- 22sinθ

=2√22cosθ

=2cosθ = RHS

Hence proved LHS = RHS

25- tanπ4+θtan3π4-θ=-1

LHS= tanπ4+θtan3π4-θ

tana+b=tana+tan⁡(b)1- tanatan⁡(b)

tana-b=tana-tan⁡(b)1+ tanatan⁡(b)

=tanπ4 +tanθ1-tanπ4tanθtan3π4-tanθ1+tan3π4tanθ

tanπ4=1 & tan3π4= -1

=1 +tanθ1-tanθ-1(1+tanθ)1-tanθ

= -1 = RHS

Hence proved LHS = RHS

26- sinπ4+θcosπ4-θ=1

Sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

LHS= sinπ4+θcosπ4-θ

=sinπ4cosθ+cosπ4sinθcosπ4cosθ+sinπ4sinθ

=√22cosθ+√22sinθ√22cosθ+√22sinθ

= 1 = RHS

Hence proved LHS = RHS

27-a) sin2A1+cos2A=tanA

LHS=sin2A1+cos2A

Sin2A= 2sinAcosA

Cos2A= cos2A – sin2A

=2sinAcosA1+cos2A-sin2A

Sin2A = 1 – cos2A

=2sinAcosA1+cos2A-(1-cos2A)

=2sinAcosA2cos2A

=2sinA2cosA

= tanA = RHS

Hence proved LHS = RHS

b) sin212θ1+cos212θ=tan12θ

LHS=sin212θ1+cos212θ

Sin212θ= 2sin12θcos12θ

Cos212θ= cos212θ – sin212θ

=2sin12θcos12θ1+cos212θ-sin212θ

Sin212θ = 1 – cos212θ

=2sin12θcos12θ1+cos212θ-(1-cos212θ)

=2sin12θcos12θ2cos212θ

=2sin12θ2cos12θ

= tan12θ = RHS

Hence proved LHS = RHS

c) ±1-cos2θ1+cos2θ=sinθ1+cos2θ

LHS=±1+cos2θ1+cos2θ

LCD: 1+cos2θ1+cos2θ

=±1-cos2θ1+cos2θ ×1+cos2θ1+cos2θ

=± 12-cos2θ21+cos2θ2

= sinθ21+cos2θ

=sinθ1+cos2θ=RHS

Hence proved LHS = RHS