khai triển hằng đẳng thức : kết quả quả còn lại 4. căn (tanB.cot B) = 4 ( vì tanB.cotB = 1 ) 

\(\left(a+b\right)^2-\left(a-b\right)^2=4ab\)

Nên ta có:

\(\left(\tan\beta+\cot\beta\right)^2-\left(\cot\beta-\tan\beta\right)^2=4\cdot\tan\beta\cdot\cot\beta=4\forall\beta\).ĐPCM

tan(x)*cot(x) = 1 với mọi x.

a, \(\widehat{B}=33^0\)

b. \(\widehat{B}=76^0\)

c, \(\widehat{B}=75^0\)

d, \(\widehat{B}=38^0\)

[kí hiệu \(^"\) là phút, mình xin lỗi do nếu đánh hẳn kí hiệu phút nó sẽ bị lỗi phông]

a) \(\sin\beta\approx0,547\Rightarrow\beta\approx33^o10^"\)

b) \(\cos\beta\approx0,238\Rightarrow\beta\approx76^o14^"\)

c) \(\tan\beta\approx3,862\Rightarrow\beta\approx75^o29^"\)

d) \(\cot\beta\approx1,295\Rightarrow\beta\approx37^o41^"\)

Cho bạn CT chung về cách bấm góc bằng máy tính cầm tay đây

gttd: giá trị tìm được

G: góc

\(\left\{{}\begin{matrix}sin^{-1}\\cos^{-1}\\tan^{-1}\end{matrix}\right.\left(gttd\right)+\left(=\right)+\left(^{o'''}\right)\Rightarrow G\)

\(tan^{-1}\left(gttd+\left(x^{-1}\right)\right)+\left(=\right)+\left(^{o''''}\right)\Rightarrow G\)

\(a,sin\beta\approx0,547\Rightarrow\beta=33^o\)

\(b,cos\beta\approx0,238\Rightarrow\beta=76^o\)

\(c,tan\beta\approx3,862\Rightarrow\beta=75^o\)

\(d,cotg\beta\approx1,295\Rightarrow\beta=38^o\)

c: 2(sin^6a+cos^6a)+1

=2[(sin^2a+cos^2a)^3-3*sin^2a*cos^2a]+1

=2-6sin^2acos^2a+1

=3-6*sin^2a*cos^2a

=3(sin^4a+cos^4a)

a:

Sửa đề: =-tana*tanb

 \(VT=\left(\dfrac{sina}{cosa}-\dfrac{sinb}{cosb}\right):\left(\dfrac{cosa}{sina}-\dfrac{cosb}{sinb}\right)\)

\(=\dfrac{sina\cdot cosb-sinb\cdot cosa}{cosa\cdot cosb}:\dfrac{cosa\cdot sinb-cosb\cdot sina}{sina\cdot sinb}\)

\(=\dfrac{sin\left(a-b\right)}{cosa\cdot cosb}\cdot\dfrac{sina\cdot sinb}{sin\left(b-a\right)}\)

\(=-tana\cdot tanb\)

=VP

a) \(sin6\alpha cot3\alpha cos6\alpha=2.sin3\alpha.cos3\alpha\dfrac{cos3\alpha}{sin3\alpha}-cos6\alpha\)
\(=2cos^23\alpha-\left(2cos^23\alpha-1\right)=1\) (Không phụ thuộc vào x).

b) \(\left[tan\left(90^o-\alpha\right)-cot\left(90^o+\alpha\right)\right]^2\)\(-\left[cot\left(180^o+\alpha\right)+cot\left(270^o+\alpha\right)\right]^2\)
\(=\left[cot\alpha+cot\left(90^o-\alpha\right)\right]^2\)\(-\left[cot\alpha+cot\left(90^o+\alpha\right)\right]^2\)
\(=\left[cot\alpha+tan\alpha\right]^2-\left[cot\alpha-tan\alpha\right]^2\)
\(=4tan\alpha cot\alpha=4\). (Không phụ thuộc vào \(\alpha\)).

c) \(\left(tan\alpha-tan\beta\right)cot\left(\alpha-\beta\right)-tan\alpha tan\beta\)
\(=\left(\dfrac{sin\alpha}{cos\alpha}-\dfrac{sin\beta}{cos\beta}\right).\dfrac{cos\left(\alpha-\beta\right)}{sin\left(\alpha-\beta\right)}-tan\alpha tan\beta\)
\(=\left(\dfrac{sin\alpha cos\beta-cos\alpha sin\beta}{cos\alpha cos\beta}\right).\dfrac{cos\left(\alpha-\beta\right)}{sin\left(\alpha-\beta\right)}\)\(-\dfrac{sin\alpha sin\beta}{cos\alpha cos\beta}\)
\(=\dfrac{sin\left(\alpha-\beta\right)}{cos\alpha cos\beta}.\dfrac{cos\left(\alpha-\beta\right)}{sin\left(\alpha-\beta\right)}-\dfrac{sin\alpha sin\beta}{cos\alpha cos\beta}\)
\(=\dfrac{cos\left(\alpha-\beta\right)}{cos\alpha cos\beta}-\dfrac{sin\alpha sin\beta}{cos\alpha cos\beta}\)
\(=\dfrac{cos\alpha cos\beta+sin\alpha sin\beta-sin\alpha sin\beta}{cos\alpha cos\beta}=\dfrac{cos\alpha cos\beta}{cos\alpha cos\beta}=1\).

Theo Viet ta có \(\left\{{}\begin{matrix}tana+tanb=p\\tana.tanb=q\end{matrix}\right.\)

\(\Rightarrow tan\left(a+b\right)=\frac{tana+tanb}{1-tana.tanb}=\frac{p}{1-q}\)

\(A=cos^2\left(a+b\right)\left[1+p.tan\left(a+b\right)+q.tan^2\left(a+b\right)\right]\)

\(A=\frac{1}{1+tan^2\left(a+b\right)}\left[1+\frac{p^2}{1-q}+\frac{q.p^2}{\left(1-q\right)^2}\right]\)

\(A=\frac{\left(1-q\right)^2}{p^2+\left(1-q\right)^2}\left(1+\frac{p^2}{\left(1-q^2\right)}\right)\)

\(A=\frac{\left(1-q^2\right)}{p^2+\left(1-q\right)^2}.\left(\frac{p^2+\left(1-q\right)^2}{\left(1-q\right)^2}\right)=1\)

a) \(\dfrac{tan\alpha-tan\beta}{cot\beta-cot\alpha}=\dfrac{\dfrac{sin\alpha}{cos\alpha}-\dfrac{sin\beta}{cos\beta}}{\dfrac{cos\beta}{sin\beta}-\dfrac{cos\alpha}{sin\alpha}}\)
\(=\dfrac{\dfrac{sin\alpha cos\beta-cos\alpha sin\beta}{cos\alpha cos\beta}}{\dfrac{cos\beta sin\alpha-cos\alpha sin\beta}{sin\beta sin\alpha}}\)
\(=\dfrac{sin\beta sin\alpha}{cos\beta cos\alpha}=tan\alpha tan\beta\).

b) \(tan100^o+\dfrac{sin530^o}{1+sin640^o}=tan100^o+\dfrac{sin170^o}{1+sin280^o}\)
\(=-cot10^o+\dfrac{sin10^o}{1-sin80^o}\)\(=\dfrac{-cos10^o}{sin10^o}+\dfrac{sin10^o}{1-cos10^o}\)
\(=\dfrac{-cos10^o+cos^210^o+sin^210^o}{sin10^o\left(1-cos10^o\right)}\) \(=\dfrac{1-cos10^o}{sin10^o\left(1-cos10^o\right)}=\dfrac{1}{sin10^o}\) .

c) \(2\left(sin^6\alpha+cos^6\alpha\right)+1=2\left(sin^2\alpha+cos^2\alpha\right)\)\(\left(sin^4\alpha-sin^2\alpha cos^2\alpha+cos^4\alpha\right)+1\)
\(=2\left(sin^4\alpha+cos^4\alpha-sin^2\alpha cos^2\alpha\right)+1\)
\(=2\left(sin^4\alpha+cos^4\alpha\right)+sin^2\alpha-sin^2\alpha cos^2\alpha+\)\(cos^2\alpha-sin^2\alpha cos^2\alpha\)
\(=2\left(sin^4\alpha+cos^4\alpha\right)+sin^2\alpha\left(1-cos^2\alpha\right)+\)\(cos^2\alpha\left(1-sin^2\alpha\right)\)
\(=2\left(sin^4\alpha+cos^4\alpha\right)+sin^2\alpha.sin^2\alpha+cos^2\alpha.cos^2\alpha\)
\(=3\left(sin^4\alpha+cos^4\alpha\right)\).