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cos square alpha minus sin square alpha equals to tan square beta to prove cos square theta minus sin square theta equals to tan square alpha​

Answer»

{cos}^{2} \alpha - {sin}^{2} \alpha = {tan}^{2} \beta

To prove

{cos}^{2} \alpha \: - { \sin }^{2} \alpha = {tan}^{2} theta

Answer:

Proved if Cos²θ - Sin²θ = tan²α then Cos²α - Sin²α = tan²θ

Step-by-step EXPLANATION:

if Cos²θ - Sin²θ = tan²α

Then

Cos²α - Sin²α = tan²θ

LHS = Cos²α - Sin²α

= Cos²α(1 - Tan²α)

= (1/Sec²α) (1 - Tan²α)

Sec²α = 1 + Tan²α

= (1/(1 + Tan²α))(1 - Tan²α)

= (1 - Tan²α)/(1 + Tan²α)

putting VALUE of Tan²α

= (1 - (Cos²θ - Sin²θ))/(1 + Cos²θ - Sin²θ)

= (1 - Cos²θ + Sin²θ))/(1  - Sin²θ + Cos²θ)

USING 1 - Cos²θ = Sin²θ & 1  - Sin²θ = Cos²θ

= (Sin²θ + Sin²θ))/(Cos²θ + Cos²θ)

= 2Sin²θ/2Cos²θ

= tan²θ

= RHS

Proved


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