CM\(\tan^2-\sin^2=\tan^2\sin^2\)
Tính
A=sin^2 1 độ + sin ^2 2 độ + sin^2 3 độ +..........+ sin ^2 89 độ
B =tan 1 độ * tan 2 độ * tan 3 độ *............ tan 89độ
6. CM đẳng thức
a) \(\dfrac{sin^3\alpha+cos^3\alpha}{sin\alpha+cos\alpha}=1-sin\alpha.cos\alpha\)
c) sin4α + cos4α - sin6α - cos6α = sin2α . cos2α
b) \(\dfrac{sin^2\alpha-cos^2\alpha}{1+2sin\alpha.cos\alpha}=\dfrac{tan\alpha-1}{tan\alpha+1}\)
a: \(VT=\dfrac{\left(sina+cosa\right)^3-3\cdot sina\cdot cosa\left(sina+cosa\right)}{sina+cosa}\)
=(sina+cosa)^2-3*sina*cosa
=sin^2a+cos^2a-sina*cosa
=1-sina*cosa=VP
c: VT=(sin^2a+cos^2a)^2-2*sin^2a*cos^2a-(sin^2a+cos^2a)^3+3*sin^2a*cos^2a*(sin^2a+cos^2a)
=1-2sin^2a*cos^2a-1+3*sin^2a*cos^2a
=sin^2a*cos^2a=VP
cho tam giác ABC .chứng minh
\(sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}+sin\frac{B}{2}cos\frac{C}{2}cos\frac{A}{2}+sin\frac{C}{2}cos\frac{A}{2}cos\frac{B}{2}=sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}+tan\frac{A}{2}tan\frac{B}{2}+tan\frac{B}{2}tan\frac{C}{2}+tan\frac{C}{2}tan\frac{A}{2}\)
Tự chứng minh từng cái này rồi suy ra cái đó nhé b.
Ta có: \(sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}-sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=sin^2\frac{A}{2}\)
Tương tự ta suy ra:
\(sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}+cos\frac{A}{2}sin\frac{B}{2}cos\frac{C}{2}+cos\frac{A}{2}cos\frac{B}{2}sin\frac{C}{2}=sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}+3sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}\left(1\right)\)
Tiếp theo chứng minh:
\(2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=\frac{cosA+cosB+cosC-1}{2}\left(2\right)\)
\(sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}=\frac{3}{2}-\frac{cosA+cosB+cosC}{2}\left(3\right)\)
\(tan\frac{A}{2}tan\frac{B}{2}+tan\frac{B}{2}tan\frac{C}{2}+tan\frac{C}{2}tan\frac{A}{2}=1\left(4\right)\)
Từ (1), (2), (3), (4) suy được điều phải chứng minh
trinh le na
cho bạn 4 năm nữa cũng chưa hiểu đâu
cứuuuuuuu
\(a) A = a %2 sin 90 ^∘ + b ^2 cos 90 ^∘ + c ^2 cos 180 ^∘\)
\(b) B = 3 − sin ^2 90 ^∘ + 2 cos ^2 60 ^∘ − 3 tan ^2 45 ^∘\)
\(c) C = sin ^2 45 ^∘ − 2 sin ^2 50 ^∘ + 3 cos ^2 45 ^∘ − 2 sin ^2 40 ^∘ + 4 tan 55 ^∘ ⋅ tan 35 ^∘\)
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khó z tui chưa học mà :)
cm các đẳng thức:
a) \(\frac{1+\sin^2\alpha}{1-\sin^2\alpha}=1+2\tan^2\alpha\)
b) \(\frac{\cos\alpha}{1+\sin\alpha}+\tan\alpha=\frac{1}{\cos\alpha}\)
c) \(\frac{\sin\alpha}{1+\cos\alpha}+\frac{1+\cos\alpha}{\sin\alpha}=\frac{2}{\sin\alpha}\)
\(\frac{1+sin^2a}{1-sin^2a}=\frac{1+sin^2a}{cos^2a}=\frac{1}{cos^2a}+\frac{sin^2a}{cos^2a}=1+tan^2a+tan^2a=1+2tan^2a\)
\(\frac{cosa}{1+sina}+tana=\frac{cosa}{1+sina}+\frac{sina}{cosa}=\frac{cos^2a+sina+sin^2a}{cosa\left(1+sina\right)}=\frac{1+sina}{cosa\left(1+sina\right)}=\frac{1}{cosa}\)
\(\frac{sina}{1+cosa}+\frac{1+cosa}{sina}=\frac{sin^2a+cos^2a+2cosa+1}{\left(1+cosa\right)sina}=\frac{2+2cosa}{\left(1+cosa\right)sina}=\frac{2\left(1+cosa\right)}{\left(1+cosa\right)sina}=\frac{2}{sina}\)
S=sin(1o58′59'')3√tan(1o)+sin(2o57′58'')3√tan(2o)+sin(3o56′57'')3√tan(3o)+...+sin(57o2′3'')3√tan(57o)+sin(58o1′2'')3√tan(58o)
CM đẳng thức
a) cos4α - sin4α = 2cos2α - 1
b) \(\dfrac{cos^2\alpha+tan^2\alpha-1}{sin^2\alpha}=tan^2\alpha\)
\(a,cos^4a-sin^4a=2cos^2a-1\\ VT=\left(cos^2a-sin^2a\right)\left(cos^2a+sin^2a\right)\\ =cos^2a-sin^2a\\ =cos2a=2cos^2a-1\)
\(b,VT=\dfrac{cos^2a+\dfrac{sin^2a}{cos^2a}-1}{sin^2a}\\ =\dfrac{\dfrac{cos^4a+sin^2a-cos^2a}{cos^2a}}{sin^2a}\\ =\dfrac{\dfrac{cos^4a+\left(1-cos^2a\right)-cos^2a}{cos^2a}}{sin^2a}\\ =\dfrac{\dfrac{cos^4a+1-2cos^2a}{cos^2a}}{sin^2a}\\ =\dfrac{\dfrac{\left(1-cos^2a\right)^2}{cos^2a}}{sin^2a}\\ =\dfrac{sin^4a}{cos^2a}:sin^2a\\ =\dfrac{sin^4a}{cos^2a}\times\dfrac{1}{sin^2a}\\ =\dfrac{sin^2a}{cos^2a}=tan^2a\)
Cho A, B, C là 3 góc trong tam giác. Chứng minh rằng:
1, sin A + sin B - sin C = 4sin\(\dfrac{A}{2}\) sin \(\dfrac{B}{2}\)sin \(\dfrac{C}{2}\)
2, \(\dfrac{sinA+sinB-sinC}{cosA+cosB-cosC+1}=tan\dfrac{A}{2}tan\dfrac{B}{2}tan\dfrac{C}{2}\) (ΔABC nhọn)
3, \(\dfrac{cosA+cosB+cosC+3}{sinA+sinB+sinC}=tan\dfrac{A}{2}+tan\dfrac{B}{2}+tan\dfrac{C}{2}\)
GIÚP MÌNH VỚI!!!
1.
\(sinA+sinB-sinC=2sin\dfrac{A+B}{2}.cos\dfrac{A-B}{2}-sin\left(A+B\right)\)
\(=2sin\dfrac{A+B}{2}.cos\dfrac{A-B}{2}-2sin\dfrac{A+B}{2}.cos\dfrac{A+B}{2}\)
\(=2sin\dfrac{A+B}{2}.\left(cos\dfrac{A-B}{2}-cos\dfrac{A+B}{2}\right)\)
\(=2sin\dfrac{A+B}{2}.2sin\dfrac{A}{2}.sin\dfrac{B}{2}\)
\(=4sin\dfrac{A}{2}.sin\dfrac{B}{2}.cos\dfrac{C}{2}\)
Sao t lại đc như này v, ai check hộ phát
1. Tính giá trị biểu thức
S= cos70 +cos50 -cos10
2. Cho a+b=π/4. Cm
(1+tanα).(1+tanβ) =2
3. Tính giá trị biểu thức
P= sin^2 10¤ +sin^2 50¤ +sin^2 70¤
1.
\(cos70+cos50=2cos\dfrac{70+50}{2}.cos\dfrac{70-50}{2}=2.cos60.cos10=2.\dfrac{1}{2}cos10\)
\(cos70+cos50-cos10=0\)
2.\(tan\left(a+b\right)=\dfrac{tana+tanb}{1-tana.tanb}=1\Rightarrow tana+tanb+tana.tanb+1=2\Leftrightarrow\left(1+tana\right)\left(1+tanb\right)=2\)