Question

If tan =

sin cos

1− 2

, show that tan ( − ) = (1 – n) tan .​

Answer 1

Answer:

tanβ=

tanβ= 1−nsin

tanβ= 1−nsin 2

tanβ= 1−nsin 2 α

tanβ= 1−nsin 2 αnsinαcosα

tanβ= 1−nsin 2 αnsinαcosα

tanβ= 1−nsin 2 αnsinαcosα =

tanβ= 1−nsin 2 αnsinαcosα = sec

tanβ= 1−nsin 2 αnsinαcosα = sec 2

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 α

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα =

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 α

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα =

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 α

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)=

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ =

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1−

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα−

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan 2

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan 2 α

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan 2 αntanα

tanβ= 1−nsin 2 αnsinαcosα = sec 2 α−ntan 2 αntanα = 1+tan 2 α−ntan 2 αntanα = 1+(1−n)tan 2 αntanα ∴tan(α−β)= 1−tanαtanβtanα−tanβ = 1− 1−tanαtanβtanαntanβ tanα− 1+(1−n)tan 2 αntanα =(1−n)tanα