Question

solve without using trigonometric tables
(tan 20 / cot 70)+(cot 50 / tan 40 ) +((sin2 20 + sin2 70 )/ sin theta cos (90 - theta ) +cos theta sin (90 - theta)

Answer 1

$\displaystyle\frac{tan\,20}{cot\,70}+\frac{cot\,50}{tan\,40}+\frac{sin^220+sin^270}{sin\theta\;cos(90-\theta)+cos\theta\;sin(90-\theta)}$

$\text{using}$

$\boxed{cot\theta=tan(90-\theta)\implies\,cot\,70=tan\,20\;\&\;cot\,50=tan\,40}$

$\boxed{\bf\,sin\theta=cos(90-\theta)\implies\,sin\,70=cos\,20}$

$\boxed{\bf\,sin\theta=cos(90-\theta)\;\;\&\;\;cos\theta=sin(90-\theta)}$

$=\displaystyle\frac{tan\,20}{tan\,20}+\frac{tan\,40}{tan\,40}+\frac{sin^220+cos^270}{sin\theta\;sin\theta+cos\theta\;cos\theta}$

$=\displaystyle\,1+1+\frac{sin^220+cos^270}{sin^2\theta+cos^2\theta}$

$\text{using}$

$\boxed{\bf\,sin^2A+cos^2A=1}$

$=1+1+\frac{1}{1}$

$=3$

$\implies\boxed{\bf\frac{tan\,20}{cot\,70}+\frac{cot\,50}{tan\,40}+\frac{sin^220+sin^270}{sin\theta\;cos(90-\theta)+cos\theta\;sin(90-\theta)}=3}$