Question

How to find the cos theta if we have tan theta and sec theta?

Answer 1

tanθ+secθ=3

We rewrite this in terms of cosine:

sinθcosθ+1cosθ=3

sinθ+1cosθ=3

sinθ+1=3cosθ

There are probably other ways to go about this, but I chose to square the equation and use sin2θ=1−cos2θ:

sin2θ+2sinθ+1=9cos2θ

sin2θ+2sinθ+1−9=9cos2θ−9

sin2θ+2sinθ−8=9(cos2θ−1)

sin2θ+2sinθ−8=9(−sin2)θ

10sin2θ+2sinθ−8=0

Now, just factor.

2(5sin2θ+sinθ−4)=0

2((5sin2θ+5sinθ)+(−4sinθ−4))=0

2(5sinθ−4)(sinθ+1)=0

sinθ=−1,45

Now that we have the value for sine, we can go back to an equation we had earlier to solve for cosine:

sinθ+1=3cosθ

sinθ+13=cosθ

−1+13=cosθ=0

45+13=cosθ=35

So, cosθ=0,35

However, if cosθ=0, then tanθ and secθ are undefined. Thus, the only solution is cosθ=35.
Hope it is correct