1. ## Trigonometric identities

Prove each identity using a t-chart.

Left hand side Right hand side

a) csc2 x(1 - cos2 x)=1

b) (cot x + tan x)/sec x = csc x

2. ## Re: Trigonometric identities

Originally Posted by alejandro
Prove each identity using a t-chart.

Left hand side Right hand side

a) csc2 x(1 - cos2 x)=1

b) (cot x + tan x)/sec x = csc x
ok ... have you made the requisite t-charts to verify each identity?

3. ## Re: Trigonometric identities

could you clarify as to what do you mean by " using t-charts". Does it mean pluging in values of trigonometric ratios in terms of the sides of a right triangle like, adjacent, opposite & hypotenuse etc??

4. ## Re: Trigonometric identities

Hello, alejandro!

I'm not familiar with a t-chart.

$\text{(a) Prove: }\:\csc^2x(1-\cos^2x) \:=\:1$

Since $\csc x = \tfrac{1}{\sin x},\,\text{then: }\:\csc^2\!x = \tfrac{1}{\sin^2\!x}$

Since $\sin^2\!x + \cos^2\!x \:=\:1,\,\text{ then: }\:1 - \cos^2\!x \:=\:\sin^2\!x$

The left side becomes: . $\tfrac{1}{\sin^2\!x}\cdot\sin^2\!x \;=\;1$

$\text{(b) Prove: }\:\frac{\cot x + \tan x}{\sec x} \:=\:\csc x$

We have: . $\frac{\dfrac{\cos x}{\sin x} + \dfrac{\sin x}{\cos x}}{\dfrac{1}{\cos x}}$

Multiply by $\frac{\sin x\cos x}{\sin x\cos x}:$

. . $\frac{\sin x\cos x\left(\dfrac{\cos x}{\sin x} + \dfrac{\sin x}{\cos x}\right)}{\sin x\cos x \left(\dfrac{1}{\cos x}\right)} \;=\;\frac{\overbrace{\cos^2\!x + \sin^2\!x}^{\text{This is 1}}}{\sin x} \;=\;\frac{1}{\sin x} \;=\;\csc x$

5. ## Re: Trigonometric identities

for exercise a)csc2 x(1 - cos2 x)=1 I have to prove csc2 x(1 - cos2 x)=1
1 = 1

6. ## Re: Trigonometric identities

Originally Posted by alejandro
for exercise a)csc2 x(1 - cos2 x)=1 I have to prove csc2 x(1 - cos2 x)=1
1 = 1
Soroban completed that proof in post #4 ... did you not see it?

7. ## Re: Trigonometric identities

Originally Posted by alejandro
Prove each identity using a t-chart.

Left hand side Right hand side

a) csc2 x(1 - cos2 x)=1

b) (cot x + tan x)/sec x = csc x
We can also do it alternatively as shown