Introduction to...

If $$\displaystyle \frac{\sin x}{a}=\frac{\cos x}{b}=\frac{\tan x}{c}=k$$, then
$$bc+\displaystyle \frac{1}{ck}+\frac{ak}{1+bk}$$ is

If $$\text{cosec}\,\theta-\sin\theta=a^{3},\ \sec\theta-\cos\theta=b^{3}$$
then $$a^{2}b^{2}(a^{2}+b^{2})=$$

$$\cos\theta +\cos^{2}\theta =1$$ and $$a\sin^{12}\theta +b\sin^{10}\theta +c\sin^{8}\theta +d\sin^{6}\theta =1.$$ Then $$\displaystyle \frac{b+c}{a+d}=$$?

If $$a \cos^{3}\alpha+3a\cos\alpha.\sin^{2}\alpha=m$$ and $$a \sin^{3}\alpha+3a\cos^{2}\alpha.\sin\alpha=n$$ then, $$(m+n)^{2/3}+(m-n)^{2/3}$$ is equal to:

If $$\displaystyle \frac { \sin { \alpha  }  }{ \sin { \beta  }  } =\frac { \sqrt { 3 }  }{ 2 } $$ and $$\displaystyle \frac { \cos { \alpha  }  }{ \cos { \beta  }  } =\frac { \sqrt { 5 }  }{ 2 } ,0<\alpha ,\beta <\frac { \pi  }{ 2 } $$, then

lf $$a \sin \theta + b \cos \theta = c$$, then $$\dfrac {a - b \tan \theta}{b + a \tan \theta} =$$

 lf $$x=\displaystyle \frac{\sin^{3}p}{\cos^{2}p}, y=\displaystyle \frac{\cos^{3}p}{\sin^{2}p}$$ and $$\displaystyle \sin p+\cos p=\frac{1}{2}$$, then $$x+y=$$

If $$\displaystyle a^{2}\cos^{4}\: \theta -b^{2}\sin^{4}\, \theta =0$$, then $$\displaystyle \: \frac{\sin^{8}\, \theta }{a^{3}}+\frac{\cos^{8}\theta }{b^{3}}$$ is 

If $$\sin x+\sin ^{2}x=1$$,then the value of $$\cos ^{12}x+3\cos ^{10}x+3\cos ^{8}x+\cos ^{6}x-2$$ is equal to