Question

If sec theta = 5/4 ,verify that tan theta /1+tan^2 theta = sin theta /sec theta​

Answer 1

Sec theta =5/4

We know that,

Sec theta=hypotenuse/base.

So , imagine a right angled triangle.

A right angled triangle has Hypotenuse, Base and Perpendicular.

So, we are getting it's hypotenuse and base from sec theta (which is given here).

Hypotenuse =5

Base =4

Now, we will apply Pythagoras theorem

Hypotenuse^2=Perpendicular^2+base^2

5^2=4^2+Perpendicular^2

Perpendicular^2=5^2-4^2

Perpendicular^2=25-16

Perpendicular^2=9

Perpendicular=root 9

Perpendicular =3

tan theta=3/4

Sin theta=3/5

Therefore,

L.H.S

tan theta/1+tan^2theta

=(3/4)/1+(3/4)^2

=(3/4)/1+9/16

=(3/4)+16+9/16

=3/4÷25/16

=12/25

Now,

R.H.S

sin theta/sec theta

=(3/5)÷(5/4)

=12/25

Therefore, R.H.S=L.H.S

Hence, proved

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