\(a=\left(\frac{sina+\frac{sina}{cosa}}{cosa+1}\right)^2+1=\left(\frac{sina\left(cosa+1\right)}{cosa\left(cosa+1\right)}\right)^2+1\)

\(=tan^2a+1=\frac{1}{cos^2a}\)

\(b=\frac{sina}{cosa}\left(\frac{1+cos^2a-sin^2a}{sina}\right)=\frac{sina}{cosa}\left(\frac{2cos^2a}{sina}\right)=2cosa\)

\(c=1-\frac{cos^2a}{cot^2a}+\frac{sina.cosa}{\frac{cosa}{sina}}=1-cos^2a.\frac{sin^2a}{cos^2a}+\frac{sin^2a.cosa}{cosa}\)

\(=1-sin^2a+sin^2a=1\)

=\(\frac{cosa}{sina}\)+\(\frac{sina}{cosa+1}\)

=\(\frac{cos^2a+cosa+sin^2a}{sina\left(cosa+1\right)}\)

=\(\frac{1+cosa}{sina\left(cosa+1\right)}\)

=\(\frac{1}{sina}\)

a) Ta có:  \(\left\{ \begin{array}{l}\sin {100^o} = \sin \left( {{{180}^o} - {{80}^o}} \right) = \sin {80^o}\\\cos {164^o} = \cos \left( {{{180}^o} - {{16}^o}} \right) =  - \cos {16^o}\end{array} \right.\)

\( \Rightarrow \sin {100^o} + \sin {80^o} + \cos {16^o} + \cos {164^o}\)\( = \sin {80^o} + \sin {80^o} + \cos {16^o}-\cos {16^o}\)\( = 2\sin {80^o}.\)

b) 

Ta có:

\(\left\{ \begin{array}{l}\sin \left( {{{180}^o} - \alpha } \right) = \sin \alpha \\\cos \left( {{{180}^o} - \alpha } \right) =  - \cos \alpha \\\tan \left( {{{180}^o} - \alpha } \right) =  - \tan \alpha \\\cot \left( {{{180}^o} - \alpha } \right) =  - \cot \alpha \end{array} \right.\quad ({0^o} < \alpha  < {90^o})\)\( \Rightarrow 2\sin \left( {{{180}^o} - \alpha } \right).\cot \alpha  - \cos \left( {{{180}^o} - \alpha } \right).\tan \alpha .\cot \left( {{{180}^o} - \alpha } \right)\) \( = 2\sin \alpha .\cot \alpha  - \left( { - \cos \alpha } \right).\tan \alpha .\left( { - \cot \alpha } \right)\)\( = 2\sin \alpha .\cot \alpha  - \cos \alpha .\tan \alpha .\cot \alpha \)

\( = 2\sin \alpha .\frac{{\cos \alpha }}{{\sin \alpha }} - \cos \alpha .\left( {\tan \alpha .\cot \alpha } \right)\)\( = 2\cos \alpha  - \cos \alpha .1 = \cos \alpha .\)

a, = \(\sin^2\alpha+2\sin\alpha.\cos\alpha+\cos^2\alpha\)+ \(\sin^2\alpha-2\sin\alpha\cos\alpha+\cos^2\alpha\)

= \(2\sin^2\alpha+2\cos^2\alpha\)= 4

b,=\(\sin\alpha\cos\alpha\)(\(\frac{\sin\alpha}{\cos\alpha}+\frac{\cos\alpha}{\sin\alpha}\))

= \(\sin\alpha\cos\alpha.\frac{\sin^2\alpha+\cos^2\alpha}{\sin\alpha\cos\alpha}\)

=1

#mã mã#

b, =1 

mk viết thiếu

#mã mã#

a) \(\left(sin\alpha+cos\alpha\right)^2+\left(sin\alpha-cos\alpha\right)^2\)

\(=sin^2\alpha+2sin\alpha\cdot cos\alpha+cos^2\alpha+sin^2\alpha-2sin\alpha\cdot cos\alpha+cos^2\alpha\)

\(=2\left(sin^2\alpha+cos^2\alpha\right)\)

\(=2\)

b) Vẽ hình minh họa cho dễ nhìn nè :

A B C α

\(sin\alpha\cdot cos\alpha\cdot\left(tan\alpha+cot\alpha\right)\)

\(=\frac{AC}{BC}\cdot\frac{AB}{BC}\cdot\left(\frac{AC}{AB}+\frac{AB}{AC}\right)\)

\(=\frac{AC\cdot AB\cdot AC}{BC\cdot BC\cdot AB}+\frac{AC\cdot AB\cdot AB}{BC\cdot BC\cdot AC}\)

\(=\left(\frac{AC}{BC}\right)^2+\left(\frac{AB}{BC}\right)^2\)

\(=sin^2\text{α}+cos^2\text{α}\)

\(=1\)

a) ta có : \(sin\alpha.cos\alpha\left(tan\alpha+cot\alpha\right)=sin\alpha.cos\alpha\left(\dfrac{sin\alpha}{cos\alpha}+\dfrac{cos\alpha}{sin\alpha}\right)\)

\(=sin^2\alpha+cos^2\alpha=1\)

b) ta có : \(\left(sin^2\alpha+cos^2\alpha\right)^2+\left(sin\alpha-cos\alpha\right)^2\)

\(=1^2+1-2sin\alpha.cos=2\left(1-2sin\alpha.cos\alpha\right)\)

c) ta có : \(tan^2\alpha-sin^2\alpha.tan^2\alpha=tan^2\alpha\left(1-sin^2\alpha\right)\)

\(=\dfrac{sin^2\alpha}{cos^2\alpha}.cos^2\alpha=sin^2\alpha\)

\(\frac{1-cosa}{1-cos^2a}-\frac{1}{1+cosa}=\frac{1-cosa}{\left(1-cosa\right)\left(1+cosa\right)}-\frac{1}{1+cosa}=\frac{1}{1+cosa}-\frac{1}{1+cosa}=0\)

\(\frac{1-sin^2a.cos^2a}{cos^2a}-cos^2a=\frac{1}{cos^2a}-\frac{sin^2a.cos^2a}{cos^2a}-cos^2a\)

\(=\frac{1}{cos^2a}-\left(sin^2a+cos^2a\right)=\frac{1}{cos^2a}-1\)

\(=\frac{1-cos^2a}{cos^2a}=\frac{sin^2a}{cos^2a}=tan^2a\)