a.
\(\dfrac{sina}{1+cosa}+\dfrac{1+cosa}{sina}=\dfrac{sin^2a+\left(1+cosa\right)^2}{sina\left(1+cosa\right)}=\dfrac{sin^2a+cos^2a+2cosa+1}{sina\left(1+cosa\right)}\)
\(=\dfrac{2cosa+2}{sina\left(1+cosa\right)}=\dfrac{2\left(1+cosa\right)}{sina\left(1+cosa\right)}=\dfrac{2}{sina}\)
b.
\(\dfrac{tan^2a-sin^2a}{cot^2a-cos^2a}=\dfrac{\dfrac{sin^2a}{cos^2a}-sin^2a}{\dfrac{cos^2a}{sin^2a}-cos^2a}=\dfrac{sin^2a\left(\dfrac{1}{cos^2a}-1\right)}{cos^2a\left(\dfrac{1}{sin^2a}-1\right)}=\dfrac{\dfrac{sin^2a\left(1-cos^2a\right)}{cos^2a}}{\dfrac{cos^2a\left(1-sin^2a\right)}{sin^2a}}=\dfrac{sin^6a}{cos^6a}=tan^6a\)
c.
\(\dfrac{1}{tana+cota}=\dfrac{1}{\dfrac{sina}{cosa}+\dfrac{cosa}{sina}}=\dfrac{sina.cosa}{sin^2a+cos^2a}=sina.cosa\)
d.
\(sin^2a.tan^2a+4sin^2a-tan^2a+3cos^2a\)
\(=tan^2a\left(sin^2a-1\right)+sin^2a+3\left(sin^2a+cos^2a\right)\)
\(=-tan^2a.cos^2a+sin^2a+3\)
\(=-\dfrac{sin^2a}{cos^2a}.cos^2a+sin^2a+3\)
\(=-sin^2a+sin^2a+3=3\)
3.
\(\dfrac{sin^2a}{sina-cosa}-\dfrac{sina+cosa}{tan^2a-1}=\dfrac{sin^2a}{sina-cosa}-\dfrac{sina+cosa}{\dfrac{sin^2a}{cos^2a}-1}\)
\(=\dfrac{sin^2a}{sina-cosa}-\dfrac{cos^2a\left(sina+cosa\right)}{sin^2a-cos^2a}=\dfrac{sin^2a}{sina-cosa}-\dfrac{cos^2a\left(sina+cosa\right)}{\left(sina-cosa\right)\left(sina+cosa\right)}\)
\(=\dfrac{sin^2a}{sina-cosa}-\dfrac{cos^2a}{sina-cosa}=\dfrac{sin^2a-cos^2a}{sina-cosa}\)
\(=\dfrac{\left(sina-cosa\right)\left(sina+cosa\right)}{sina-cosa}=sina+cosa\)
