Question

If tan theta = 3/4 , then cos²theta - sin²theta​

Answer 1

Question :-

If $\bf{tan\theta = \dfrac{3}{4}}$ , then find the value of $\bf{cos^{2}\theta - sin^{2}}$

To Find :-

The value of $\bf{cos^{2}\theta - sin^{2}}$.

Given :-

$\bf{tan\theta = \dfrac{3}{4}}$

We Know :-

Pythagoras theorem :-

$\underline{\boxed{\bf{h^{2} = p^{2} + b^{2}}}}$

Trigonometrical identities :-

• $\bf{tan\theta = \dfrac{p}{b}}$

• $\bf{sin\theta = \dfrac{p}{h}}$

• $\bf{cos\theta = \dfrac{b}{h}}$

Concept :-

Given the value of $\bf{tan\theta}$ is ¾ .

But we know that , $\bf{tan\theta}$ is p/b.

So from this we get :-

$\bf{tan\theta = \dfrac{p}{b} = \dfrac{3}{4}}$

Hence, from this information we get height as 3 units and base as 4 units.

So by using the Pythagoras theorem ,we can find the value of Hypotenuse.

And then by substituting the values of $\bf{sin\theta}$ and $\bf{sin\theta}$ in the Equation , we can find the required value.

Solution :-

To find the Hypotenuse :—

Given :-

• Height = 3 cm

• Base = 4 cm

By using the Pythagoras theorem and substituting the values in it , we get :-

$:\implies \bf{h^{2} = p^{2} + b^{2}} \\ \\ \\ :\implies \bf{h^{2} = 3^{2} + 4^{2}} \\ \\ \\ :\implies \bf{h = \sqrt{3^{2} + 4^{2}}} \\ \\ \\ :\implies \bf{h = \sqrt{9 + 16}} \\ \\ \\ :\implies \bf{h = \sqrt{25}} \\ \\ \\ :\implies \bf{h = 5 units} \\ \\ \\ \therefore \purple{\bf{h = 5 units}}$.

Hence, the hypotenuse is 5 units.

Hence :-

• $\bf{sin\theta = \dfrac{p}{h} = \dfrac{3}{5}}$

• $\bf{cos\theta = \dfrac{b}{h} = \dfrac{4}{5}}$

By substituting the values in the Equation , we get :-

$:\implies \bf{cos^{2}\theta - sin^{2}} \\ \\ \\ :\implies \bf{\bigg(\dfrac{4}{5}\bigg)^{2} - \bigg(\dfrac{3}{5}\bigg)^{2}} \\ \\ \\ :\implies \bf{\dfrac{16}{25} - \dfrac{9}{25}} \\ \\ \\ :\implies \bf{\dfrac{16 - 9}{25}} \\ \\ \\ :\implies \bf{\dfrac{7}{25}} \\ \\ \\ :\therefore \purple{\bf{cos^{2}\theta - sin^{2} = \dfrac{7}{25}}}$.

Hence, the value of $\bf{cos^{2}\theta - sin^{2}}$ is 7/25.

Answer 2

Step-by-step explanation:

Let theta = x

tan x=3/4. , sin x=? and. sec x=?

sec^2x=1+tan^2x=1+9/16= 25/16

sec x=+/-(5/4). Answer

cos x= +/- 4/5

sin x=+/-√(1-cos^2x) = +/- √(1–16/25)=+/- √(9/25) = +/-3/5. Answer.

Second-method:-

tan x= 3/4

or. sin x/cos x= 3/4

or. (sin x)/3= (cos x)/4 =k(led)

or. sin x= 3k. , cos x=4k

sin^2x+cos^2x=1. =>. 9k^2+16k^2=1. or. 25k^2 = 1

k = +/-1/5

sin x= 3k = +/-3/5. Answer

and. cos x=4k = +/-(4/5). or sec x= +/-(5/4). Answer.