Question

sec theta minus tan theta is equals to a + 1 / a -1 then cos theta=?

Answer 1

you can see your answer

Answer 2

Concept:

To answer this question we need the concept of trigonometry.

Trigonometry is the branch of mathematics dealing with the relations of sides of triangles using relevant functions namely; sine, cosine, tangent, etc.

Formula required= sec²$\Theta$-tan²$\Theta$=1

Given:

We are given an expression using trigonometric functions.

sec $\Theta$-tan $\Theta$=a+1/a-1

To find:

We are asked to find the trigonometric function namely cos theta

Solution:

We have to find the value of the cos $\Theta$

We are given the expression:

sec$\Theta$ - tan$\Theta$ =a+1/a-1

We know that:

sec²$\Theta$ - tan²$\Theta$=1 (formula from trigonometry)

⇒(sec $\Theta$ - tan $\Theta$)(sec $\Theta$ +tan $\Theta$)=1 (using the formula a²-b²=(a-b)(a+b))

⇒(a+1/a-1)(sec $\Theta$ +tan $\Theta$)=1

⇒(sec $\Theta$ +tan $\Theta$)=a-1/a+1

⇒sec $\Theta$ -tan $\Theta$+sec $\Theta$ +tan $\Theta$=sec $\Theta$ -tan $\Theta$+a-1/a+1    (adding sec $\Theta$ -tan $\Theta$ on both sides)

⇒2sec$\Theta$=(a+1/a-1)+(a-1/a+1) ( substituing value of sec $\Theta$ -tan $\Theta$ )

Taking L.C.M on right side we have:

⇒2sec$\Theta$=(a+1)²+(a-1)²/a²-1

⇒2sec$\Theta$=(a²+2a+1+a²-2a+1)/a²-1

⇒2sec$\Theta$=2a²+2/a²-1

⇒2sec$\Theta$=2(a²+1)/a²-1

⇒sec$\Theta$=a²+1/a²-1

⇒1/Cos$\Theta$=a²+1/a²-1 (∵sec$\Theta$=1/Cos$\Theta$)

∴Cos$\Theta$=a²-1/a²+1

Hence, our required value is Cos$\Theta$=a²-1/a²+1