Question

If cos theta =2x/1+x^2, then tan theta =​

Answer 1

Given:-

cosθ = $\frac{2x}{(1+x²)}$

To find:-

Value of tanθ in terms of x

How to find:-

In this question we are going to use pythagoras theorem and basic trigonometric ratios.

Solution:-

cosθ = $\frac{2x}{(1+x²)}$

We know that cosθ = $\frac{Base}{Hypotenuse}$

Therefore,

Base = 2x

Hypotenuse = 1 + x²

According to pythagoras theorem,

(1 + x²)² = (2x)² + (Height)²

1 + x⁴ + 2x² - 4x² = (Height)²

(Height)² = 1 + x⁴ - 2x²

(Height)² = (1 - x²)²

Therefore,

Height = 1 - x²

We know that tanθ = $\frac{Height}{Base}$

Therefore,

tanθ = $\frac{(1 - x²)}{2x}$

Important formulas related to trigonometry:-

i) sin²θ + cos²θ = 1

ii) 1 + tan²θ = sec²θ

iii) 1 + cot²θ = cosec²θ

iv) sin(A + B) = sinAcosB + cosAsinB

v) sin(A - B) = sinAcosB - cosAsinB

vi) cos(A + B) = cosAcosB - sinAsinB

vii) cos(A - B) = cosAcosB + sinAsinB

viii) tan(A + B) = $\rm \frac{tanA + tanB}{1 - tanAtanB}$

ix) tan(A - B) = $\rm \frac{tanA - tanB}{1 + tanAtanB}$

x) cot(A + B) = $\rm \frac{cotAcotB - 1}{cotA + cotB}$

xi) cot(A - B) = $\rm \frac{cotAcotB + 1}{cotA - cotB}$