Question

If cot q + cosec q = 5, find cos q.

Answer 1

Answer:

Step-by-step explanation:

The question is to find $cos q$, if $cot (q)+ cosec (q)=5$

$cot(q)+cosec(q)=5$    →(1)

This is our equation (1). We know $\frac{1}{sin(q)}=cosec(q)$ and $\frac{cos(q)}{sin(q)}=cot(q)$. Put these in equation (1). That is,

 $\frac{1}{sin(q)}+\frac{cos(q)}{sin(q)}=5$, we can write this as, $\frac{1+cos(q)}{sin(q)} =5$

Take $sin(q)$ to the other side of the equal sign, then the equation will be,

$1+cos(q)=5sin(q)$

squaring on both sides,  $(1+cos(q))^{2} =5^{2} sin^{2} (q)$

                                                $=25sin^{2} (q)$ →(2)

We know that $sin^{2}q +cos^{2}q=1,sin^{2}q=1-cos^{2}q$. Substitute this in equation (2),

∴  $(1+cos(q))^{2} =25 (1-cos^{2} q)$

We can use the identity $a^{2}-b^{2}=(a+b)(a-b)$ for $1-cos^{2} q$. Here $a=1$ and $b=cos^{2} q$

$(1+cosq)(1+cosq) =25 (1-cos q)(1+cosq)$, $(1+cosq)$ is common on both sides.Then

$(1+cosq) =25 (1-cos q)$

Rearrange the equation to get the final answer,

$(1+cosq) =25 -25cos q\\\\cosq+25cosq =25 -1\\\\cosq(1+25)=24\\\\cosq(26)=24\\\\cosq=\frac{24}{26}$

This is the final answer, $cosq=\frac{24}{26}$

Thank you