Question

Prove that
sin thrita /sin (90-thrita)+cos thrita /cos(90-thrita)= sec thrita *cosec thrita.​

Answer 1

To prove:

$\sf{\frac{sin\theta}{sin(90-\theta)}+\frac{cos\theta}{cos(90-\theta)}=sec\theta.cosec\theta}$

Proof:

$\sf{L.H.S.=\frac{sin\theta}{sin(90-\theta)}+\frac{cos\theta}{cos(90-\theta)}}$

$\sf\blue{sin(90-\theta)=cos\theta \ and \ cos(90-\theta)=sin\theta}$

$\sf{=\frac{sin\theta}{cos\theta}+\frac{cos\theta}{sin\theta}}$

$\sf{=\frac{sin^{2}\theta+cos^{2}\theta}{sin\theta.cos\theta}}$

$\sf\blue{sin^{2}\theta+cos^{2}\theta}$

$\sf{=\frac{1}{sin\theta.cos\theta}}$

$\sf{=\frac{1}{sin\theta}\times\frac{1}{cos\theta}}$

$\sf\blue{\frac{1}{sin\theta}=cosec\theta \ and \ \frac{1}{cos\theta}=sec\theta}$

$\sf{=cosec\theta.sec\theta}$

$\sf{=sec\theta.cosec\theta}$

$\sf{=R.H.S.}$

$\sf{Hence, \ proved.}$

$\sf\purple{\tt{\frac{sin\theta}{sin(90-\theta)}+\frac{cos\theta}{cos(90-\theta)}=sec\theta.cosec\theta}}$

Answer 2

Answer:

Yes it's correct

Step-by-step explanation:sin (90- theta)=cos theta

Cos (90- theta)=sin theta

sin thrita /sin (90-thrita)+cos thrita /cos(90-thrita)= sec thrita *cosec thrita.​

By taking lhs,

=Sin theta/Cos theta+Cos theta /sin theta

=Sin^2 theta+cos^2/sin theta ×Cos theta (since sin^2a+cos^2a=1)

=1/Sin theta ×Cos theta ( since 1/sina=coseca ,1/cosa=seca)

=cosec theta×sec theta

There fore Hence proved